Research ArticleAnApproach to theGeometric-Arithmetic Index forGraphsunderTransformationsrsquo Fact over Pendent Paths
Muhammad Asif 1 Hamad Almohamedh 2 Muhammad Hussain 1
Khalid M Alhamed 3 Abdulrazaq A Almutairi 4 and Sultan Almotairi 5
1Department of Mathematics COMSATS University Islamabad Lahore Campus Lahore 54000 Pakistan2King Abdulaziz City for Science and Technology (KACST) Riyadh Riyadh Saudi Arabia3IT Programs Center Faculty of IT Department Institute of Public Administration Riyadh 11141 Saudi Arabia4Information and Computer Center e Public Authority for Applied Education and Training e Ministry of EducationKuwait Kuwait5Department of Natural and Applied Sciences Faculty of Community College Majmaah UniversityMajmaah 11952 Saudi Arabia
Correspondence should be addressed to Hamad Almohamedh halmohamedhkacstedusa andSultan Almotairi almotairimuedusa
Received 10 May 2021 Accepted 12 June 2021 Published 24 June 2021
Academic Editor Muhammad Javaid
Copyright copy 2021Muhammad Asif et alampis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Graph theory is a dynamic tool for designing andmodeling of an interconnection system by a graphampe vertices of such graph areprocessor nodes and edges are the connections between these processors nodes ampe topology of a system decides its best useGeometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlyinginterconnection networks or graphs Transformation over graph is also an important tool to define new network of their ownchoice in computer science In this work we discuss transformed family of graphs Let Γkl
n be the connected graph comprises by k
number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ Let Aα(Γkln ) and A
βα(Γkl
n ) be thetransformed graphs under the fact of transformations Aα and A
βα 0le αle l 0le βle k minus 1 respectively In this work we obtained
new inequalities for the graph family Γkln and transformed graphs Aα(Γkl
n ) and Aβα(Γkl
n ) which involve GA(Γ) ampe presence ofGA(Γ) makes the inequalities more general than all those which were previously defined for the GA index Furthermore wecharacterize extremal graphs which make the inequalities tight
1 Introduction
ampe advancement in technology mainly networking com-puter biological and electrical networks made practicablethe accurate data transfer within very small duration ampeInternet social media biological ecological and neuralnetworks are few examples of such networks Telecommu-nication is based on interconnection networks which used toshare data files Similarly data exchange using computingdevices is also based on computer network through datalinkage optical fiber cable (OFC) and wireless media such asWi-Fi Different algorithms are used for directing arrang-ingdetermining numerical calculations and image
processing Multiprocessor interconnection networks(MINs) are used to design powerful microprocessors andmemory chips [1 2]
Graph theory provides a fundamental tool for designingand analyzing such networks Naturally the interconnectionsystem is modeled by the graph with processor nodes asvertices and links between these nodes as edges of suchgraph Graph theory and interconnection networks providea thorough understanding of these interrelated topicsthrough their topology ampe topology of a graph providesinformation about the manner in which vertices joined in agraph ampe topological indices are graph invariants used tostudy the topology of graphs Other than computer
HindawiComplexityVolume 2021 Article ID 3745862 13 pageshttpsdoiorg10115520213745862
networks graph theory is considered as a powerful tool indifferent areas of research such as in coding theory databasemanagement system circuit design secret sharing schemesand theoretical chemistry [3] ampe topological descriptors ofseveral interconnection networks are already been computedin [4ndash6] Along with interconnection networks these in-variants are equally important in chemical graph theorywhich deals with problems in chemistry using associatedgraph of chemical compounds [7]
ampe study of underlying substance using their graph withthe help of graph invariants plays an important role in chem-informatics pharmaceutical sciences materials scienceengineering and so forth [8 9] Among theoretical mo-lecular descriptors topological indices have an impact inchemistry due to the prediction of physio-chemical prop-erties of the underlying substance Its role in the QSPRQSAR analysis to model physical and chemical properties ofmolecules is also remarkable [10ndash12] Actually topologicalindices are designed on the ground of transformation whichassociates a numeric value with the graph which charac-terizes its topology [13] ampe first topological index namedWinner index was proposed in 1947 by Winner [14] Itprovides best correlation with the boiling points of alkanesampe discovery of the Winner index provides emerging re-search platform to the research community ampe interest inaccurate prediction of physio-chemical properties encour-aged the researchers to propose a large class of topologicalindices For the first time an index is defined on the base ofend verticesrsquo degrees of the edges by Milan Randic namedRandic connectivity index [15]
RA(Γ) 1113944uvisinE(Γ)
1degudegv
1113968 (1)
Due to this reason it has attained a great attraction of theresearchers till now In 2009 Vukicevic and Furtula [16]introduced the geometric-arithmetic index
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(2)
GA has correlation coefficient of 0972 with heat offormation of benzene hydrocarbons Also in case ofldquostandard enthalpy of vaporizationrdquo its accuracy is 9 morethan the Randic index Due to this reason GA was studiedmore than all other indices in the last decade ampe bonds andextremal characterization of graphs regarding the GA indexwere studied at some extent in [17ndash24] It encouraged us tostudy the GA index for Γkl
n and transformed graphs Aα(Γkln )
and Aβα(Γkl
n ) under the fact of transformations Aα and Aβα
0le αle l 0le βle k minus 1 respectively We characterize extremalgraphs for all of these families of graphs
2 Results and Discussion
amproughout this work let graph Γkln comprise with n-vertex
simple connected graph Γ along with k pendent paths oflength lge 2 attached with v isin Γ having degree dv ge 2 ampeorder of Γkl
n is n + kl size is m + kl anddeg1 δΓ le deg2 le deg3 le middot middot middot leΔΓ + 1 is its degree sequence
Let graph Γ Γ(V E) be with the degree of vertex u isin Γand δΓ le degu leΔΓ and δΓ le degv leΔΓ + 1 be the degrees ofv isin Γkl
n For validity of our proved results we defined thefollowing list of useful graphs
Type I let δΓ le degu leΔΓ where u isin V(Γ) Γkln of type I is
obtained by attaching pendent paths of length l with verticesof degree degu ge 2 in such a way that the vertices withpendent path are adjacent to the vertices without pendentpaths
ampe graph of type I is shown in Figure 1(a)Type II Γkl
n of type II is the graph of type I withdegu ΔΓ forallu isin V(Γ)
ampe graph of type II is shown in Figure 1(b)Before attempting the major problem we prove the
following preposition
Proposition 1 Let xge 2 then
2x(x minus 1)
1113968
2x minus 1le2
x(x + 1)
1113968
2x + 1 (3)
Proof Let xge 2
2x(x + 1)
1113968
2x + 1minus2
x(x minus 1)
1113968
2x minus 1
2
x(x + 1)
1113968(2x minus 1) minus 2
x(x minus 1)
1113968(2x + 1)
(2x + 1)(2x minus 1)
2
x
radic[x
(x + 1)
1113968+(x minus 1)
(x + 1)
1113968minus x
(x minus 1)
1113968minus (x + 1)
(x minus 1)
1113968]
(2x + 1)(2x minus 1)
2
x
radic x + 1
radicx minus
x2
minus 11113968
1113874 1113875 minusx minus 1
radicx minus
x2
minus 11113968
1113874 11138751113876 1113877
(2x + 1)(2x minus 1)
2
x
radicx minus
x2
minus 11113968
1113874 1113875(x + 1
radicminus
x minus 1
radic)
(2x + 1)(2x minus 1)ge 0
(4)
2 Complexity
ampe above calculations implies
2x(x minus 1)
1113968
2x minus 1le2
x(x + 1)
1113968
2x + 1 (5)
Theorem 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (6)
Equality holds for graphs of type II And
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (7)
Equality holds for graph of type II
Proof Let a simple graph Γ be of order n size m maximumdegree ΔΓ and minimum δΓ Γkl
n be the graph formed by k
number of paths having length l pendent at distinct verticesu isin Γ such that 2le degu le δΓ ampe geometric-arithmetic in-dex of any graph Γ is
GA(Γ) 1113944stisinE(Γ)
2degsdegt
1113968
degs + degt
(8)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + kl
and for st isinE(Γkln )(degs +degt) isin 34degu +2degu +degv1113864
degu +degv +1
ampe edge set of Γkln partitioned as
A3 st isin Γkln degs 1 degt 21113966 1113967
A4 st isin Γkln degs degt 21113966 1113967
Adegu+2 st isin Γkln δΓ le degs degu leΔΓ degt 21113966 1113967
Adegu+degv st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
(9)
and Adegu+degv+1 st isin Γkln δΓ ledegs degudegt degv +11113966
leΔΓ
Gk nlG
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
helliphelliphellip
(a)
Gk nl
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
G
hellip hellip hellip
(b)
Figure 1 Graph Γkln (a) Graph Γkl
n of type I (b) Graph Γkln of type II
Complexity 3
GA Γkln1113872 1113873 1113944
st are edges of
pendent paths
2degsdegt
1113968
degs + degt
+ 1113944st are edges of Γ
2degsdegt
1113968
degs + degt
GA Γkln1113872 1113873 1113944
stisinA3
2degsdegt
1113968
degs + degt
+ 1113944stisinA4
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+3
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
(10)
ampe construction of GA(Γkln ) implies that the cardinality
of A3 is k ie |A3| k |A4| k(l minus 2) |Adegu+3| k|Adegu+degu+1|le kΔΓ and |Adegu+degu
|le kΔΓampe function f(x)
2ax
radica + x is decreasing where ale x is a constant So for
δΓ minimum degree of vertices of Γ and maximum degreeΔΓ we have
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857ge2
2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 11138572
ΔΓ + 1( 1113857δΓ
1113969
ΔΓ + 1( 1113857 + δΓle2
degs + 1( 1113857degt
1113969
degs + 1( 1113857 + degt
2
degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk (11)
From equation (10) we have
GA Γkln1113872 1113873ge
2k2
radic
3+ k(l minus 2) +
2k2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 1113857+2ΔΓk
δΓ ΔΓ + 1( 1113857
1113969
δΓ + ΔΓ + 1minus kΔΓ + GA(Γ) (12)
After simplification we obtain
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (13)
Now again set
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857le2
2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857 1113944
stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
le2
ΔΓ + 1( 1113857ΔΓ
1113969
ΔΓ + 1( 1113857 + ΔΓ+ GA(Γ) minus
2kΔΓ
δΓΔΓ1113969
δΓ + ΔΓ
(14)
which implies from Proposition 1 and the characteristics off(x) 2
xyradic x + y in equation (10) We get the following
inequality
GA Γkln1113872 1113873le
2k2
radic
3+ k(l minus 2) +
2k2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857+2ΔΓk
ΔΓ ΔΓ + 1( 1113857
1113969
ΔΓ + ΔΓ + 1+ GA(Γ) minus
2ΔΓk
δΓΔΓ1113969
δΓ + ΔΓ (15)
4 Complexity
After simplification we obtain
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (16)
Inequalities (13) and (16) complete the proofCorollary 1 shows generalization of the above defined
inequalities One can get more inequalities of their desire byreplacing GA(Γ) with already defined bonds of the GAindex
Corollary 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ +2m
ΔΓδΓ
1113968
δΓ + ΔΓ
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + m
(17)
Equality holds for regular graph of the type II
Proof Using results of ampeorem 1 and inequality regardingthe geometric index proved in [25 26] as
2mΔΓδΓ
1113968
ΔΓ + δΓleGA(Γ)lem (18)
we get desired results
21GraphTransformations Let H(Γ) sub E(Γ) Γprime Γ minus H bethe new graph generated by removing set edges of H(Γ) andΓPrime Γ minus V1(Γ) be the new graph generated by deleting set ofvertices V1(Γ) sub V(Γ) We use the following transforma-tions as used in [27] ampese transformations have solid effectover GA of Γkl
n
Transformation A let wj isin V(Γ) degwjge 2 for
1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 comprise Γkl
n ampen
A Γkln1113872 1113873 Γkl
n minus 1113944k
j1u2ju
3j u
3ju
4j u
lminus 1j u
lj1113966 1113967
+ 1113944
k
j1wju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(19)
ampe transformation A is shown in Figure 2In ampeorem 2 we discuss the effect of transformation A
over the GA index
Theorem 2 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(20)
Equality holds for all graphs of type II
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891 +
2k2
radic
3
+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(21)
Complexity 5
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
networks graph theory is considered as a powerful tool indifferent areas of research such as in coding theory databasemanagement system circuit design secret sharing schemesand theoretical chemistry [3] ampe topological descriptors ofseveral interconnection networks are already been computedin [4ndash6] Along with interconnection networks these in-variants are equally important in chemical graph theorywhich deals with problems in chemistry using associatedgraph of chemical compounds [7]
ampe study of underlying substance using their graph withthe help of graph invariants plays an important role in chem-informatics pharmaceutical sciences materials scienceengineering and so forth [8 9] Among theoretical mo-lecular descriptors topological indices have an impact inchemistry due to the prediction of physio-chemical prop-erties of the underlying substance Its role in the QSPRQSAR analysis to model physical and chemical properties ofmolecules is also remarkable [10ndash12] Actually topologicalindices are designed on the ground of transformation whichassociates a numeric value with the graph which charac-terizes its topology [13] ampe first topological index namedWinner index was proposed in 1947 by Winner [14] Itprovides best correlation with the boiling points of alkanesampe discovery of the Winner index provides emerging re-search platform to the research community ampe interest inaccurate prediction of physio-chemical properties encour-aged the researchers to propose a large class of topologicalindices For the first time an index is defined on the base ofend verticesrsquo degrees of the edges by Milan Randic namedRandic connectivity index [15]
RA(Γ) 1113944uvisinE(Γ)
1degudegv
1113968 (1)
Due to this reason it has attained a great attraction of theresearchers till now In 2009 Vukicevic and Furtula [16]introduced the geometric-arithmetic index
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(2)
GA has correlation coefficient of 0972 with heat offormation of benzene hydrocarbons Also in case ofldquostandard enthalpy of vaporizationrdquo its accuracy is 9 morethan the Randic index Due to this reason GA was studiedmore than all other indices in the last decade ampe bonds andextremal characterization of graphs regarding the GA indexwere studied at some extent in [17ndash24] It encouraged us tostudy the GA index for Γkl
n and transformed graphs Aα(Γkln )
and Aβα(Γkl
n ) under the fact of transformations Aα and Aβα
0le αle l 0le βle k minus 1 respectively We characterize extremalgraphs for all of these families of graphs
2 Results and Discussion
amproughout this work let graph Γkln comprise with n-vertex
simple connected graph Γ along with k pendent paths oflength lge 2 attached with v isin Γ having degree dv ge 2 ampeorder of Γkl
n is n + kl size is m + kl anddeg1 δΓ le deg2 le deg3 le middot middot middot leΔΓ + 1 is its degree sequence
Let graph Γ Γ(V E) be with the degree of vertex u isin Γand δΓ le degu leΔΓ and δΓ le degv leΔΓ + 1 be the degrees ofv isin Γkl
n For validity of our proved results we defined thefollowing list of useful graphs
Type I let δΓ le degu leΔΓ where u isin V(Γ) Γkln of type I is
obtained by attaching pendent paths of length l with verticesof degree degu ge 2 in such a way that the vertices withpendent path are adjacent to the vertices without pendentpaths
ampe graph of type I is shown in Figure 1(a)Type II Γkl
n of type II is the graph of type I withdegu ΔΓ forallu isin V(Γ)
ampe graph of type II is shown in Figure 1(b)Before attempting the major problem we prove the
following preposition
Proposition 1 Let xge 2 then
2x(x minus 1)
1113968
2x minus 1le2
x(x + 1)
1113968
2x + 1 (3)
Proof Let xge 2
2x(x + 1)
1113968
2x + 1minus2
x(x minus 1)
1113968
2x minus 1
2
x(x + 1)
1113968(2x minus 1) minus 2
x(x minus 1)
1113968(2x + 1)
(2x + 1)(2x minus 1)
2
x
radic[x
(x + 1)
1113968+(x minus 1)
(x + 1)
1113968minus x
(x minus 1)
1113968minus (x + 1)
(x minus 1)
1113968]
(2x + 1)(2x minus 1)
2
x
radic x + 1
radicx minus
x2
minus 11113968
1113874 1113875 minusx minus 1
radicx minus
x2
minus 11113968
1113874 11138751113876 1113877
(2x + 1)(2x minus 1)
2
x
radicx minus
x2
minus 11113968
1113874 1113875(x + 1
radicminus
x minus 1
radic)
(2x + 1)(2x minus 1)ge 0
(4)
2 Complexity
ampe above calculations implies
2x(x minus 1)
1113968
2x minus 1le2
x(x + 1)
1113968
2x + 1 (5)
Theorem 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (6)
Equality holds for graphs of type II And
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (7)
Equality holds for graph of type II
Proof Let a simple graph Γ be of order n size m maximumdegree ΔΓ and minimum δΓ Γkl
n be the graph formed by k
number of paths having length l pendent at distinct verticesu isin Γ such that 2le degu le δΓ ampe geometric-arithmetic in-dex of any graph Γ is
GA(Γ) 1113944stisinE(Γ)
2degsdegt
1113968
degs + degt
(8)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + kl
and for st isinE(Γkln )(degs +degt) isin 34degu +2degu +degv1113864
degu +degv +1
ampe edge set of Γkln partitioned as
A3 st isin Γkln degs 1 degt 21113966 1113967
A4 st isin Γkln degs degt 21113966 1113967
Adegu+2 st isin Γkln δΓ le degs degu leΔΓ degt 21113966 1113967
Adegu+degv st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
(9)
and Adegu+degv+1 st isin Γkln δΓ ledegs degudegt degv +11113966
leΔΓ
Gk nlG
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
helliphelliphellip
(a)
Gk nl
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
G
hellip hellip hellip
(b)
Figure 1 Graph Γkln (a) Graph Γkl
n of type I (b) Graph Γkln of type II
Complexity 3
GA Γkln1113872 1113873 1113944
st are edges of
pendent paths
2degsdegt
1113968
degs + degt
+ 1113944st are edges of Γ
2degsdegt
1113968
degs + degt
GA Γkln1113872 1113873 1113944
stisinA3
2degsdegt
1113968
degs + degt
+ 1113944stisinA4
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+3
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
(10)
ampe construction of GA(Γkln ) implies that the cardinality
of A3 is k ie |A3| k |A4| k(l minus 2) |Adegu+3| k|Adegu+degu+1|le kΔΓ and |Adegu+degu
|le kΔΓampe function f(x)
2ax
radica + x is decreasing where ale x is a constant So for
δΓ minimum degree of vertices of Γ and maximum degreeΔΓ we have
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857ge2
2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 11138572
ΔΓ + 1( 1113857δΓ
1113969
ΔΓ + 1( 1113857 + δΓle2
degs + 1( 1113857degt
1113969
degs + 1( 1113857 + degt
2
degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk (11)
From equation (10) we have
GA Γkln1113872 1113873ge
2k2
radic
3+ k(l minus 2) +
2k2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 1113857+2ΔΓk
δΓ ΔΓ + 1( 1113857
1113969
δΓ + ΔΓ + 1minus kΔΓ + GA(Γ) (12)
After simplification we obtain
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (13)
Now again set
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857le2
2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857 1113944
stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
le2
ΔΓ + 1( 1113857ΔΓ
1113969
ΔΓ + 1( 1113857 + ΔΓ+ GA(Γ) minus
2kΔΓ
δΓΔΓ1113969
δΓ + ΔΓ
(14)
which implies from Proposition 1 and the characteristics off(x) 2
xyradic x + y in equation (10) We get the following
inequality
GA Γkln1113872 1113873le
2k2
radic
3+ k(l minus 2) +
2k2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857+2ΔΓk
ΔΓ ΔΓ + 1( 1113857
1113969
ΔΓ + ΔΓ + 1+ GA(Γ) minus
2ΔΓk
δΓΔΓ1113969
δΓ + ΔΓ (15)
4 Complexity
After simplification we obtain
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (16)
Inequalities (13) and (16) complete the proofCorollary 1 shows generalization of the above defined
inequalities One can get more inequalities of their desire byreplacing GA(Γ) with already defined bonds of the GAindex
Corollary 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ +2m
ΔΓδΓ
1113968
δΓ + ΔΓ
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + m
(17)
Equality holds for regular graph of the type II
Proof Using results of ampeorem 1 and inequality regardingthe geometric index proved in [25 26] as
2mΔΓδΓ
1113968
ΔΓ + δΓleGA(Γ)lem (18)
we get desired results
21GraphTransformations Let H(Γ) sub E(Γ) Γprime Γ minus H bethe new graph generated by removing set edges of H(Γ) andΓPrime Γ minus V1(Γ) be the new graph generated by deleting set ofvertices V1(Γ) sub V(Γ) We use the following transforma-tions as used in [27] ampese transformations have solid effectover GA of Γkl
n
Transformation A let wj isin V(Γ) degwjge 2 for
1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 comprise Γkl
n ampen
A Γkln1113872 1113873 Γkl
n minus 1113944k
j1u2ju
3j u
3ju
4j u
lminus 1j u
lj1113966 1113967
+ 1113944
k
j1wju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(19)
ampe transformation A is shown in Figure 2In ampeorem 2 we discuss the effect of transformation A
over the GA index
Theorem 2 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(20)
Equality holds for all graphs of type II
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891 +
2k2
radic
3
+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(21)
Complexity 5
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
ampe above calculations implies
2x(x minus 1)
1113968
2x minus 1le2
x(x + 1)
1113968
2x + 1 (5)
Theorem 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (6)
Equality holds for graphs of type II And
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (7)
Equality holds for graph of type II
Proof Let a simple graph Γ be of order n size m maximumdegree ΔΓ and minimum δΓ Γkl
n be the graph formed by k
number of paths having length l pendent at distinct verticesu isin Γ such that 2le degu le δΓ ampe geometric-arithmetic in-dex of any graph Γ is
GA(Γ) 1113944stisinE(Γ)
2degsdegt
1113968
degs + degt
(8)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + kl
and for st isinE(Γkln )(degs +degt) isin 34degu +2degu +degv1113864
degu +degv +1
ampe edge set of Γkln partitioned as
A3 st isin Γkln degs 1 degt 21113966 1113967
A4 st isin Γkln degs degt 21113966 1113967
Adegu+2 st isin Γkln δΓ le degs degu leΔΓ degt 21113966 1113967
Adegu+degv st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
(9)
and Adegu+degv+1 st isin Γkln δΓ ledegs degudegt degv +11113966
leΔΓ
Gk nlG
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
helliphelliphellip
(a)
Gk nl
ul3
u33
u23
u13
w3 w2 w1
u32
u22
u12
u31
u21
u11
ul3ndash1
ul3ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
G
hellip hellip hellip
(b)
Figure 1 Graph Γkln (a) Graph Γkl
n of type I (b) Graph Γkln of type II
Complexity 3
GA Γkln1113872 1113873 1113944
st are edges of
pendent paths
2degsdegt
1113968
degs + degt
+ 1113944st are edges of Γ
2degsdegt
1113968
degs + degt
GA Γkln1113872 1113873 1113944
stisinA3
2degsdegt
1113968
degs + degt
+ 1113944stisinA4
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+3
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
(10)
ampe construction of GA(Γkln ) implies that the cardinality
of A3 is k ie |A3| k |A4| k(l minus 2) |Adegu+3| k|Adegu+degu+1|le kΔΓ and |Adegu+degu
|le kΔΓampe function f(x)
2ax
radica + x is decreasing where ale x is a constant So for
δΓ minimum degree of vertices of Γ and maximum degreeΔΓ we have
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857ge2
2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 11138572
ΔΓ + 1( 1113857δΓ
1113969
ΔΓ + 1( 1113857 + δΓle2
degs + 1( 1113857degt
1113969
degs + 1( 1113857 + degt
2
degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk (11)
From equation (10) we have
GA Γkln1113872 1113873ge
2k2
radic
3+ k(l minus 2) +
2k2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 1113857+2ΔΓk
δΓ ΔΓ + 1( 1113857
1113969
δΓ + ΔΓ + 1minus kΔΓ + GA(Γ) (12)
After simplification we obtain
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (13)
Now again set
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857le2
2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857 1113944
stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
le2
ΔΓ + 1( 1113857ΔΓ
1113969
ΔΓ + 1( 1113857 + ΔΓ+ GA(Γ) minus
2kΔΓ
δΓΔΓ1113969
δΓ + ΔΓ
(14)
which implies from Proposition 1 and the characteristics off(x) 2
xyradic x + y in equation (10) We get the following
inequality
GA Γkln1113872 1113873le
2k2
radic
3+ k(l minus 2) +
2k2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857+2ΔΓk
ΔΓ ΔΓ + 1( 1113857
1113969
ΔΓ + ΔΓ + 1+ GA(Γ) minus
2ΔΓk
δΓΔΓ1113969
δΓ + ΔΓ (15)
4 Complexity
After simplification we obtain
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (16)
Inequalities (13) and (16) complete the proofCorollary 1 shows generalization of the above defined
inequalities One can get more inequalities of their desire byreplacing GA(Γ) with already defined bonds of the GAindex
Corollary 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ +2m
ΔΓδΓ
1113968
δΓ + ΔΓ
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + m
(17)
Equality holds for regular graph of the type II
Proof Using results of ampeorem 1 and inequality regardingthe geometric index proved in [25 26] as
2mΔΓδΓ
1113968
ΔΓ + δΓleGA(Γ)lem (18)
we get desired results
21GraphTransformations Let H(Γ) sub E(Γ) Γprime Γ minus H bethe new graph generated by removing set edges of H(Γ) andΓPrime Γ minus V1(Γ) be the new graph generated by deleting set ofvertices V1(Γ) sub V(Γ) We use the following transforma-tions as used in [27] ampese transformations have solid effectover GA of Γkl
n
Transformation A let wj isin V(Γ) degwjge 2 for
1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 comprise Γkl
n ampen
A Γkln1113872 1113873 Γkl
n minus 1113944k
j1u2ju
3j u
3ju
4j u
lminus 1j u
lj1113966 1113967
+ 1113944
k
j1wju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(19)
ampe transformation A is shown in Figure 2In ampeorem 2 we discuss the effect of transformation A
over the GA index
Theorem 2 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(20)
Equality holds for all graphs of type II
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891 +
2k2
radic
3
+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(21)
Complexity 5
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
GA Γkln1113872 1113873 1113944
st are edges of
pendent paths
2degsdegt
1113968
degs + degt
+ 1113944st are edges of Γ
2degsdegt
1113968
degs + degt
GA Γkln1113872 1113873 1113944
stisinA3
2degsdegt
1113968
degs + degt
+ 1113944stisinA4
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+3
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
(10)
ampe construction of GA(Γkln ) implies that the cardinality
of A3 is k ie |A3| k |A4| k(l minus 2) |Adegu+3| k|Adegu+degu+1|le kΔΓ and |Adegu+degu
|le kΔΓampe function f(x)
2ax
radica + x is decreasing where ale x is a constant So for
δΓ minimum degree of vertices of Γ and maximum degreeΔΓ we have
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857ge2
2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 11138572
ΔΓ + 1( 1113857δΓ
1113969
ΔΓ + 1( 1113857 + δΓle2
degs + 1( 1113857degt
1113969
degs + 1( 1113857 + degt
2
degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk (11)
From equation (10) we have
GA Γkln1113872 1113873ge
2k2
radic
3+ k(l minus 2) +
2k2lowast ΔΓ + 1( 1113857
1113969
2 + ΔΓ + 1( 1113857+2ΔΓk
δΓ ΔΓ + 1( 1113857
1113969
δΓ + ΔΓ + 1minus kΔΓ + GA(Γ) (12)
After simplification we obtain
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (13)
Now again set
22lowast degs + 1( 1113857
1113969
2 + degs + 1( 1113857le2
2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857 1113944
stisinAdegu+degv+1
2degsdegt
1113968
degs + degt
+ 1113944stisinAdegu+degv
2degsdegt
1113968
degs + degt
le2
ΔΓ + 1( 1113857ΔΓ
1113969
ΔΓ + 1( 1113857 + ΔΓ+ GA(Γ) minus
2kΔΓ
δΓΔΓ1113969
δΓ + ΔΓ
(14)
which implies from Proposition 1 and the characteristics off(x) 2
xyradic x + y in equation (10) We get the following
inequality
GA Γkln1113872 1113873le
2k2
radic
3+ k(l minus 2) +
2k2lowast δΓ + 1( 1113857
1113969
2 + δΓ + 1( 1113857+2ΔΓk
ΔΓ ΔΓ + 1( 1113857
1113969
ΔΓ + ΔΓ + 1+ GA(Γ) minus
2ΔΓk
δΓΔΓ1113969
δΓ + ΔΓ (15)
4 Complexity
After simplification we obtain
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (16)
Inequalities (13) and (16) complete the proofCorollary 1 shows generalization of the above defined
inequalities One can get more inequalities of their desire byreplacing GA(Γ) with already defined bonds of the GAindex
Corollary 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ +2m
ΔΓδΓ
1113968
δΓ + ΔΓ
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + m
(17)
Equality holds for regular graph of the type II
Proof Using results of ampeorem 1 and inequality regardingthe geometric index proved in [25 26] as
2mΔΓδΓ
1113968
ΔΓ + δΓleGA(Γ)lem (18)
we get desired results
21GraphTransformations Let H(Γ) sub E(Γ) Γprime Γ minus H bethe new graph generated by removing set edges of H(Γ) andΓPrime Γ minus V1(Γ) be the new graph generated by deleting set ofvertices V1(Γ) sub V(Γ) We use the following transforma-tions as used in [27] ampese transformations have solid effectover GA of Γkl
n
Transformation A let wj isin V(Γ) degwjge 2 for
1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 comprise Γkl
n ampen
A Γkln1113872 1113873 Γkl
n minus 1113944k
j1u2ju
3j u
3ju
4j u
lminus 1j u
lj1113966 1113967
+ 1113944
k
j1wju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(19)
ampe transformation A is shown in Figure 2In ampeorem 2 we discuss the effect of transformation A
over the GA index
Theorem 2 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(20)
Equality holds for all graphs of type II
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891 +
2k2
radic
3
+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(21)
Complexity 5
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
After simplification we obtain
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + GA(Γ) (16)
Inequalities (13) and (16) complete the proofCorollary 1 shows generalization of the above defined
inequalities One can get more inequalities of their desire byreplacing GA(Γ) with already defined bonds of the GAindex
Corollary 1 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ΔΓ + 1and minimum δΓ en
GA Γkln1113872 1113873ge k
3 l minus ΔΓ minus 2( 1113857 + 22
radic
3+ 2
ΔΓ + 1
11139682
radic
ΔΓ + 3( 1113857+ΔΓ
δΓ1113969
ΔΓ + δΓ + 1( 1113857⎛⎜⎜⎝ ⎞⎟⎟⎠⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦ +2m
ΔΓδΓ
1113968
δΓ + ΔΓ
GA Γkln1113872 1113873le k
22
radic+ 3l minus 63
+2
2 δΓ + 1( 1113857
1113969
δΓ + 3+ 2kΔ32Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
δΓ + ΔΓ1113888 1113889
⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ + m
(17)
Equality holds for regular graph of the type II
Proof Using results of ampeorem 1 and inequality regardingthe geometric index proved in [25 26] as
2mΔΓδΓ
1113968
ΔΓ + δΓleGA(Γ)lem (18)
we get desired results
21GraphTransformations Let H(Γ) sub E(Γ) Γprime Γ minus H bethe new graph generated by removing set edges of H(Γ) andΓPrime Γ minus V1(Γ) be the new graph generated by deleting set ofvertices V1(Γ) sub V(Γ) We use the following transforma-tions as used in [27] ampese transformations have solid effectover GA of Γkl
n
Transformation A let wj isin V(Γ) degwjge 2 for
1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 comprise Γkl
n ampen
A Γkln1113872 1113873 Γkl
n minus 1113944k
j1u2ju
3j u
3ju
4j u
lminus 1j u
lj1113966 1113967
+ 1113944
k
j1wju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(19)
ampe transformation A is shown in Figure 2In ampeorem 2 we discuss the effect of transformation A
over the GA index
Theorem 2 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(20)
Equality holds for all graphs of type II
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891 +
2k2
radic
3
+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(21)
Complexity 5
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
Equality holds for all graphs of the type II and α 0
Proof Let a simple graph Γ be of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(22)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klAfter successive applications of transformation A as Aααle l minus 1 the edge set of Aα(Γkl
E3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs 1 degt 21113966 1113967
E4 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n degs degt 21113966 1113967
Edegu+α+2 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 11113966 1113967
Edegu+α+3 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + α + 1leΔΓ + α + 1 degt 21113966 1113967
Edegu+α+1 Aα Γkln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu + αleΔΓ degt 11113966 1113967
Edegu+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degs degu degt degv leΔΓ1113966 1113967
Edegu+α+1+degvAα Γ
kln1113872 11138731113872 1113873 st isin Γkl
n δΓ le degv degu leΔΓ degs degu + α + 1 degt degv1113966 1113967
(23)
GA Aα Γkln1113872 11138731113872 1113873 1113944
E degs+degt( ) Aα Γkln( )( )
subeE Aα Γkln( )( )
1113944
stisinE degs+degt( ) Aα Γkl
n( )( )
11139442
degsdegt
1113968
degs + degt
(24)
AA (Гnkl)
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2 ul1
ul1ndash1
ul1ndash2u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12
u32
u22
u12
w2w3
w1
wk
w2w3
w1
wk
u31
u21
u11
u31
u21
u11
hellip
helliphellip
hellip
hellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Гnkl
Figure 2 Transformation A
6 Complexity
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
ampe cardinality of A3 is k ie |E3(Aα(Γkln ))| k ||E4(Aα
(Γkln ))| k(l minus α minus 2) |Edegu+α+2(Aα(Γkl
n ))| kα and|Edegu+α+3(Aα(Γkl
n ))| k ampe function f(x) 2ax
radica + x
is decreasing where ale x is a constant So for δΓ minimumdegree of Γ and ΔΓ maximum for any graph
22lowast degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857ge2
2lowast ΔΓ + 1 + α( 1113857
1113969
2 + ΔΓ + 1 + α( 1113857
2ΔΓ + α + 1( 1113857δΓ
1113969
ΔΓ + α + 1( 1113857 + δΓle2
degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus ΔΓk
(25)
Substituting these changes in equation (24) we have thefollowing inequality
GA Aα Γkln1113872 11138731113872 1113873ge
2k2
radic
3+ k(l minus 2 minus α)
+2αk
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2k
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2kΔΓ
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+ GA(Γ) minus ΔΓK
(26)
After simplification we get the required result
GA Aα Γkln1113872 11138731113872 1113873ge 2k
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + 1 + α( 1113857 + δΓ⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
+2k
2
radic
3+ GA(Γ) + kl minus k 2 + α + ΔΓ( 1113857
(27)
Now again from equation (24) and inequalities
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
geGA(Γ) minus2ΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
2degs + α + 1
1113968
1 + degs + α + 1( 1113857le
2δΓ + α + 1
1113968
1 + δΓ + α + 1( 1113857
2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓge 1113944
stisinAdegu+degv+α+1
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
22lowast degs + α + 1( 1113857
1113969
2 + degs + α + 1( 1113857le2
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
(28)
GA Aα Γkln1113872 11138731113872 1113873le
2k2
radic
3+2αk
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857
+2k
2lowast δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857+2ΔΓk
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+ k(l minus 2 minus α) + GA(Γ) minus2kΔΓ
ΔΓδΓ1113969
ΔΓ + δΓ
(29)
After simplification we obtain
GA Aα Γkln1113872 11138731113872 1113873le 2k
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113890 1113891
+2k
2
radic
3+ k(l minus 2 minus α) + GA(Γ) + 2kΔ33Γ
middot
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113890 1113891
(30)Inequalities (27) and (30) complete the proofTransformation B let wj isin V(Γ) degwj
ge 2 for1le jle kle n and paths pendent at wj of the formwju
1j u1
ju2j u2
ju3j ulminus 1
j ulj1113966 1113967 which comprises Γkl
n ampenfor fixed vertex w1
B Γkln1113872 1113873 Γkl
n minus u1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
+ w1u1j u
1ju
2j u
2ju
3j u
lminus 1j u
lj1113966 1113967
(31)
ampe transformation B is shown in Figure 3 and Aβα shown
in Figure 4Transformation A
βα let 0le αle l minus 1 and 0le βle k minus 1 ampe
transformation Aβα is the composition of successive applica-
tions of transformationA andB asAα andBβ respectively [27]Inampeorem 3 we discuss the effect of transformation A
βα
over the GA index
Theorem 3 Let graph Γkln comprise of n-vertex simple
connected graph Γ along with k pendent paths of length lge 2attached with v isin Γ of degree dv ge 2 maximum degree ofv isin Γkl
n is ΔΓ + 1 and minimum δΓ en
Complexity 7
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
ΔΓ + α + 1
1113968 αΔΓ + α + 2
+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠
+ 2
ΔΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)+ k(l minus 2 minus α) minus ΔΓ(k minus β) +
2k2
radic
3+ GA(Γ)
BB (Гn
kl)Гnkl
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2 ul
2
ul2ndash1
ul2ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
u33
u23
u13
u33
u23
u13
u3k
u2k
u1k
u3k
u2k
u1k
u32
u22
u12 u3
2
u22
u12w2
w3 w1
wk
w2
w3 w1
wk
u31
u21
u11
u31
u21
u11hellip
helliphellip
helliphellip
hellip
helliphellip
hellip
hellip
hellip
hellip
Г
Г
Figure 3 Transformation B for fixed vertex w1
Гnkl A1
1(Гnkl)
A21(Гn
kl)
ulk
u33
u23
u13
u33
u23
u13 u3
3
u23
u13
u3k
u2k
u1k
u3k u3
ku2
k u2k
u1k
u1k
w3w2 w1
wk
w3w2
w1 w3w2
w1wk wk
u32
u22
u12
u32 u3
2u22 u2
2
u31
u21
u11
u31
u21
u12
u11
u12
u31
u21
u11
u11
A11
A21
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ulk
ulkndash1
ulkndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul2
ul2ndash1
ul2ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul3
ul3ndash1
ul3ndash2
ul1
ul1ndash1
ul1ndash2
ul1
ul1ndash1
ul1ndash2
ul1ndash1
ul1
ul1ndash1
ul1ndash2
hellip
hellip
hellip
hellip
hellip
hellip
hellip
hellip
helliphellip
hellip
hellip helliphellip
hellip helliphellip
hellip
Figure 4 Transformation Aβα
8 Complexity
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(32)
Equality holds for graph of the type II with α 0 andβ 0
Proof Let a simple graph Γ of order n size m minimumdegree δΓ and maximum ΔΓ Let Γkl
n be the graph formed byk number of paths of length l pendent at distinct fullyconnected vertices of Γ ampe geometric-arithmetic index ofany graph Γ is
GA(Γ) 1113944stisinE(G)
2degsdegt
1113968
degs + degt
(33)
ampe construction of Γkln lge 2 implies |E(Γkl
n )| m + klLet ulowast be the fixed vertex Applications of transformation A
βα
has an effect over the edge set partition as E(degs+degt)(A
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Substituting these changes in equation (35) we obtainedthe following inequality
GA Aβα Γ
kln1113872 11138731113872 1113873ge
2k2
radic
3+2α(k minus β minus 1)
1lowast ΔΓ + α + 1( 1113857
1113969
1 + ΔΓ + α + 1( 1113857+2(k minus β minus 1)
2 ΔΓ + α + 1( 1113857
1113969
2 + ΔΓ + α + 1( 1113857
+2α(β + 1)
1lowast ΔΓ +(β + 1)(1 + α)( 1113857
1113969
1 + ΔΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 ΔΓ +(β + 1)(1 + α)( 1113857
1113969
2 + ΔΓ +(β + 1)(1 + α)( 1113857
+2ΔΓ(k minus β minus 1)
ΔΓ + α + 1( 1113857δΓ1113969
ΔΓ + α + 1( 1113857 + δΓ+2ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
ΔΓ +(β + 1)(1 + α)( 1113857 + δΓ
+ GA(Γ) + k(l minus 2 minus α) minus ΔΓ(k minus β)
(39)
10 Complexity
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873ge 2(k minus β minus 1)
δΓ + α + 11113969 α
ΔΓ + α + 2+
2
radic
ΔΓ + α + 3+ΔΓ
δΓ1113969
ΔΓ + α + 1 + δΓ⎛⎜⎜⎝ ⎞⎟⎟⎠ + 2
ΔΓ +(β + 1)(1 + α)
1113969
timesα(β + 1)
1 + ΔΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + ΔΓ +(β + 1)(1 + α)1113888 1113889 + 2
ΔΓ
ΔΓ +(β + 1)(1 + α)( 1113857δΓ1113969
δΓ + ΔΓ +(β + 1)(1 + α)
+ k(l minus 2 minus α) minus ΔΓ(k minus β) +2k
2
radic
3+ GA(Γ)
(40)
Now again substituting the following inequalities inequation (35)
22 degs + 1 + α( 1113857
1113969
2 + degs + 1 + α( 1113857le2
2 δΓ + 1 + α( 1113857
1113969
2 + δΓ + 1 + α( 1113857
2degs + α + 1( 1113857degt
1113969
degs + α + 1( 1113857 + degt
le2
δΓ + α + 1( 1113857
1113969
δΓ + α + 1( 1113857 + 1
22 degs +(β + 1)(1 + α)( 1113857
1113969
2 + degs +(β + 1)(1 + α)( 1113857le2
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857
2degs +(β + 1)(1 + α)( 1113857
1113969
1 + degs +(β + 1)(1 + α)( 1113857le2
δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
1113944stisinAdegu+degv
2degs + degt
1113968
degs + degt
leGA(Γ) +2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(41)
Complexity 11
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
We obtain
GA Aβα Γ
kln1113872 11138731113872 1113873le
2k2
radic
3+2α(k minus β minus 1)
1lowast δΓ + α + 1( 1113857
1113969
1 + δΓ + α + 1( 1113857+2(k minus β minus 1)
2 δΓ + α + 1( 1113857
1113969
2 + δΓ + α + 1( 1113857
+2α(β + 1)
1lowast δΓ +(β + 1)(1 + α)( 1113857
1113969
1 + δΓ +(β + 1)(1 + α)( 1113857
+2(β + 1)
2 δΓ +(β + 1)(1 + α)( 1113857
1113969
2 + δΓ +(β + 1)(1 + α)( 1113857+ GA(Γ) +
2ΔΓ(k minus β minus 1)
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓ
+2ΔΓ
ΔΓ + 1( 1113857ΔΓ1113969
ΔΓ + 1( 1113857 + ΔΓminus2ΔΓ(k minus β)
ΔΓδΓ1113969
ΔΓ + δΓ
(42)
After simplification we get the required result
GA Aβα Γ
kln1113872 11138731113872 1113873le 2(k minus β minus 1)
δΓ + α + 11113969 α
δΓ + α + 2+
2
radic
δΓ + α + 31113888 1113889 + k(l minus 2 minus α) +
2k2
radic
3
+ 2
δΓ +(β + 1)(1 + α)
1113969 α(β + 1)
1 + δΓ +(β + 1)(1 + α)+
2
radic(β + 1)
2 + δΓ +(β + 1)(1 + α)1113888 1113889
+ 2ΔΓΔΓ
1113968(k minus β)
ΔΓ + 1
1113968
2ΔΓ + 1minus
δΓ
1113968
ΔΓ + δΓ1113888 1113889 + GA(Γ)
(43)
Inequalities (40) and (43) complete the proof
3 Conclusion
ampe study of mathematical aspect regarding topological indicesis a partially open problem [20 28 29] For which memberfamily of graphs the certain index has a minimal or maximalvalue In this work we discussed for this fundamental questiongeneral graphs with pendent paths for the most studied indexnamed geometric-arithmetic index GA and developed tightbounds by characterizing graphs In ampeorems 2 and 3 for thefirst time we defined tight bonds for the transformed graphsunder the effect of transformations defined in [27]
Data Availability
No data were used to support this study
Disclosure
ampis paper has not been published elsewhere and it will notbe submitted anywhere else for publication
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors have equally contributed to the study
Acknowledgments
ampe authors extend their appreciation to the deputyship forResearch and Innovation Ministry of Education in SaudiArabia for funding this research work through the projectno IFP-2020-17
References
[1] L N Bhuyan Q Yang and D P Agrawal ldquoPerformance ofmultiprocessor interconnection networksrdquo Computer vol 22no 2 pp 25ndash37 1989
[2] T y Feng ldquoA survey of interconnection networksrdquo Com-puter vol 14 no 12 pp 12ndash27 1981
[3] J A Bondy and U S R Murty Graph eory with Appli-cations Macmillan London UK 1976
[4] M Imran S Hayat and M Y H Mailk ldquoOn topologicalindices of certain interconnection networksrdquo AppliedMathematics and Computation vol 244 pp 936ndash951 2014
[5] M Imran A Q Baig and H Ali ldquoOn topological propertiesof dominating David derived networksrdquo Canadian Journal ofChemistry vol 94 no 2 pp 137ndash148 2016
12 Complexity
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
Complexity 13
[6] M Imran M A Iqbal Y Liu A Q Baig W Khalid andM A Zaighum ldquoComputing eccentricity-based topologicalindices of 2-power interconnection networksrdquo Journal ofChemistry vol 2020 Article ID 3794592 2020
[7] N Trinajstic Chemical Graph eory Routledge EnglandUK 2018
[8] J-B Liu H Shaker I Nadeem and M Hussain ldquoTopologicalaspects of Boron nanotubesrdquo Advances in Materials Scienceand Engineering vol 2018 Article ID 5729291 2018
[9] M Asif M Hussain H Almohamedh et al ldquoStudy of carbonnanocones via connection zagreb indicesrdquo MathematicalProblems in Engineering vol 2021 Article ID 5539904 2021
[10] A A Dobrynin R Entringer and I Gutman ldquoWiener indexof trees theory and applicationsrdquo Acta Applicandae Mathe-maticae vol 66 no 3 pp 211ndash249 2001
[11] G Rucker and C Rucker ldquoOn topological indices boilingpoints and cycloalkanesrdquo Journal of Chemical Informationand Computer Sciences vol 39 no 5 pp 788ndash802 1999
[12] S Mondal A Dey N De and A Pal ldquoQSPR analysis of somenovel neighbourhood degree-based topological descriptorsrdquoComplex and Intelligent Systems vol 7 no 2 pp 977ndash9962021
[13] H Ahmad M Hussain W Nazeer and Y M Chu ldquoDistancebased invariants of zigzag polyhex nanotuberdquo MathematicalMethods in the Applied Sciences vol 2021 pp 1ndash21
[14] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947
[15] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975
[16] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometrical and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009
[17] Y Yuan B Zhou and N Trinajstic ldquoOn geometric-arithmeticindexrdquo Journal of Mathematical Chemistry vol 47 no 2pp 833ndash841 2010
[18] K C Das and N Trinajstic ldquoComparison between firstgeometric-arithmetic index and atom-bond connectivity in-dexrdquo Chemical Physics Letters vol 497 no 1ndash3 pp 149ndash1512010
[19] H Hua and S Zhang ldquoA unified approach to extremal treeswith respect to geometric-arithmetic Szeged and edge Szegedindicesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 65 pp 691ndash704 2011
[20] T Divnic M Milivojevic and L Pavlovic ldquoExtremal graphsfor the geometricndasharithmetic index with given minimumdegreerdquo Discrete Applied Mathematics vol 162 pp 386ndash3902014
[21] J Rodrıguez and J Sigarreta ldquoOn the geometricndasharithmeticindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 74 pp 103ndash120 2015
[22] J M Sigarreta ldquoBounds for the geometric-arithmetic index ofa graphrdquo Miskolc Mathematical Notes vol 16 no 2pp 1199ndash1212 2015
[23] A Portilla J Rodrıguez and J Sigaretta ldquoRecent lowerbounds for geometric-arithmetic indexrdquo Discrete Mathe-matics Letters vol 1 pp 59ndash82 2019
[24] R Hasni and N HM Husin ldquoBicyclic graphs with maximumgeometric-arithmetic indexrdquo Applied Mathematics E-Notesvol 20 pp 8ndash32 2020
[25] K C Das ldquoOn geometricndasharithmetic index of graphsrdquoMATCH Communications in Mathematical and in ComputerChemistry vol 64 pp 619ndash630 2010
[26] K C Das I Gutman and B Furtula ldquoSurvey on geometric-arithmetic indices of graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 65 pp 595ndash644 2011
[27] M Asif M Hussain H Almohamedh K M Alhamed andS Almotairi ldquoAn approach to the extremal inverse degreeindex for families of graphs with transformation effectrdquoJournal of Chemistry vol 2021 Article ID 6657039 2021
[28] A Ali and D Dimitrov ldquoOn the extremal graphs with respectto bond incident degree indicesrdquo Discrete Applied Mathe-matics vol 238 pp 32ndash40 2018
[29] A Ali L Zhong and I Gutman ldquoHarmonic index and itsgeneralizations extremal results and boundsrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 81 pp 249ndash311 2019
