Research Article The Wiener Index of Circulant...

Research ArticleThe Wiener Index of Circulant Graphs

Houqing Zhou

Department of Mathematics Shaoyang University Hunan 422004 China

Correspondence should be addressed to Houqing Zhou zhouhq2004163com

Received 7 February 2014 Accepted 24 March 2014 Published 9 April 2014

Academic Editor Maria N D S Cordeiro

Copyright copy 2014 Houqing Zhou This 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

Circulant graphs are an important class of interconnection networks in parallel and distributed computing In this paper we discussthe relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix andreciprocal distance matrix of circulants We obtain the following consequence 119882120582 = 119867120583 2119882119899 = 120582 2119867119899 = 120583 where WH denote theWiener index and the Harary index and 120582 120583 denote the largest eigenvalues of distance matrix and reciprocal distancematrix of circulant graphs respectively Moreover we also discuss the Wiener index of nonregular graphs with cut edges

1 Introduction

A circulant graph is a graph whose adjacency matrix (withrespect to a suitable vertex indexing) can be constructedfrom its first row by a process of continued rotation ofentries The interest of circulant graphs in graph theory andapplications has grown during the last two decades theyappeared in coding theory VLSI design Ramsey theoryand other areas Recently there is vast research on theinterconnection schemes based on circulant topology Circu-lant graphs represent an important class of interconnectionnetworks in parallel and distributed computing [1]

We consider simple graphs Let 119866 be a connected graphwith the vertex-set 119881(119866) = V

1 V2 V

119899 The distance

matrix119863 of 119866 is an 119899 times 119899matrix (119863119894119895) such that119863

119894119895is just the

distance (ie the number of edges of a shortest path) betweenthe vertices V

119894and V119895in 119866 The reciprocal distance matrixD119903

119894119895

of 119866 is also called the Harary matrix [2]For V119894isin 119881(119866) 119873

119866(V119894) denotes the set of its neighbors

in 119866 Let 120582(119866) and 120583(119866) be respectively the maximumeigenvalues of D and D119903 the distance spectral radius of 119866 isthe largest119863-eigenvalue 120582(119866) Ivanciuc et al [3] proposed touse the maximum eigenvalues of distance-based matrices asstructural descriptors they have shown that 120582(119866) and 120583(119866)are able to produce fair QSPR models for the boiling pointsmolar heat capacities vaporization enthalpies refractiveindices and densities for C

6ndashC10

alkanes

Recall the Hosoya definition of the Wiener index [4] andthe Harary index

119882(119866) = sum

119894lt119895

D119894119895 119867 (119866) = sum

119894lt119895

D119903119894119895 (1)

Since the distance matrix and related matrices based ongraph-theoretical distance are rich sources of many graphinvariants (topological indices) that have been found to beused in structure-property-activity modeling [5] it is ofinterest to study spectra and polynomials of these matrices

The Harary index 119867 = 119867(119866) of a molecular graph 119866with 119899 vertices is based on the concept of reciprocal distanceand is defined in parallel with the Wiener index as the half-sum of the off-diagonal elements of the reciprocal moleculardistance matrixD119903

119894119895= D119903(119866)

119867 =1

2

119873

sum

119894=1

119873

sum

119895=1

[D119903]119894119895 (2)

The reciprocal distancematrixD119903119894119895can be simply obtained

by replacing all off-diagonal elements of the distance matrixD119894119895by their reciprocals

D119903119894119895=

1

D119894119895

(3)

It should be noted that diagonal elements (D119903)119894119894are all equal

to zero by definition This matrix was first mentioned by

Hindawi Publishing CorporationJournal of ChemistryVolume 2014 Article ID 742121 4 pageshttpdxdoiorg1011552014742121

2 Journal of Chemistry

Balaban et al [6] The maximum eigenvalues of variousmatrices have recently attracted attention of mathematicalchemists [6ndash8]

The Harary index and the related indices such as itsextension to heterosystems [7] and the hyper-Harary index[8] have shown a modest success in structure-property cor-relations [9] but the use of these indices in combination withother descriptors appears to be very efficacious in improv-ing the QSPR (quantitative structure-property relationship)models

The paper is organized as follows In Section 2 we bringforward an interesting phenomenon In Section 3 we analyzethe cause of this phenomenon to emerge we report ourresults for the maximum eigenvalues of the Wiener matrixand the Harary matrix of 119903-circulant graphs Finally inSection 4 we discuss theWiener index of nonregular graphs

2 An Interesting Phenomenon

Recall that for a positive integer 119899 and set 119878 sube 0 1 2 119899 minus1 the circulant graph 119866(119899 119878) is the graph with 119899 verticeslabeled with integers modulo 119899 such that each vertex 119894 isadjacent to 119878 other vertices 119894 + 119904 (mod 119899) | 119904 isin 119878 Theset 119878 is called a symbol of 119866(119899 119878) As we will consider onlyundirected graphs without loops we assume that 0 notin 119878 and119904 isin 119878 if and only if 119899 minus 119904 isin 119878 and therefore the vertex 119894 isadjacent to vertices 119894 plusmn 119904 (mod 119899) for each 119904 isin 119878 In otherwords a graph is circulant if it is Cayley graph on the circulantgroup that is its adjacency matrix is circulant

For any circulant graph 119866 it is a regular graph A 119903-circulant graph is called regular of degree (or valency) 119903when every vertex has precisely 119903 neighbors Let 119889 denotethe diameter of 119866 then 119889 is less than the number of distincteigenvalues of the adjacency matrix of 119866 (see [10])

Now consider 4-circulant graphs with 6 7 18 ver-tices respectively By a straightforwardmathematical calcula-tion we obtain somedata on theWiener index119882 theHararyindex119867 and the eigenvalues 120582 120583 of the Wiener matrix andthe Harary matrix as follows in Table 1

From Table 1 it is easy to observe that119882119867 and 120582120583 areequal Moreover we have 2119882119899 = 120582 and 2119867119899 = 120583

We again observe Table 2From Table 2 we also obtain the same results previously

when the number of vertices is fixed in 119866(119899 119878)Why does this phenomenon occur We will be interested

in trying to explain something about this phenomenon Toprove our result we need a few more lemmas

Lemma 1 (see [11]) Let B = (B119894119895) be an 119899 times 119899 nonnegative

irreducible symmetric matrix (119899 ge 2) with row sums1198611 1198612 119861

119899 If 120588(B) is the maximum eigenvalue of B then

radicsum119899

119894=11198612

119894

119899le 120588 (B) le max

1le119894le119899

119899

sum

119895=1

B119894119895radic119861119895

119861119894

(4)

Table 1

119899 119882 119867 119889 120582 120583 119882119867 120582120583 2119882119899 2119867119899

6 18 135 2 6 45 43 43 6 457 28 175 2 8 5 85 85 8 58 40 22 2 10 55 2011 2011 10 559 54 27 2 12 6 2 2 12 610 75 953 3 15 193 4519 4519 15 19311 99 1103 3 18 203 2710 2710 18 20312 126 42 3 21 7 3 3 21 713 156 1433 3 24 223 3611 3611 24 22314 196 63712 4 28 9112 4813 4813 28 911215 240 2354 4 32 476 19247 19247 32 47616 288 1943 4 36 9712 43297 43297 36 971217 340 4256 4 40 253 245 245 40 25318 405 3845 5 45 12815 675128 675128 45 12815

with either equality if and only if 1198611= 1198612= sdot sdot sdot = 119861

119899or there

is a permutation matrix Q such that

Q119879BQ = (0 CC119879 0) (5)

where all the row sums of C are equal

Lemma 2 (see [12]) Let 119866 be a r-regular graph of order 119899If 119903 120582

2 120582

119899are the adjacency eigenvalues of 119866 then its D-

eigenvalues are 2119899minus119903minus2 andminus(120582119894+2) 119894 = 2 3 119899 where D-

eigenvalues denote the eigenvalues of the distancematrix119863(119866)

Lemma 3 (see [13]) Let 119866 be a r-circulant graph on 119899 verticesand the Wiener index 119882 of 119866 then the following equalityholds

119882 ge 119899 sdot (119899 minus 1) minus |119864| (6)

where |119864| denotes the number of edges

Lemma 4 (see [14]) Let 1198621be a component of 119866 minus 119906 and

V1 V2 V

119896(1 le 119896 le deg

119866(119906)minusdeg

1198621

(119906)) isin 119873119866(119906)119873

1198621

(119906)If 1198731198621

(119906) V = 1198731198621

(V) where V is adjacent to 119906 in 1198621

let 1198661015840 be obtained by removing an edge 119906V119904of 119866 and add a

new edge VV119904(1 le 119904 le 119896) For all 1 le 119904 le 119896 if a vertex

119908 isin 119881(119866) 119881(1198621) cup 119906 satisfies dist

119866(119908 V119904) lt dist

1198661015840(119908 V119904)

then 120582(119866) lt 120582(1198661015840) where dist

119866(119906 V) denotes the distance

between a pair of vertices 119906 V in 119866

3 Main Results

In this section we will prove the following two theorems

Theorem 5 Let119866 be a r-circulant graph on 119899 (119899 ge 4) verticesthen the following equality holds

119882(119866)

119867 (119866)=120582 (119866)

120583 (119866) (7)

Journal of Chemistry 3

Table 2

Graph 119878 119882 119867 120582 120583 119882119867 120582120583 2119882119899 2119867119899

119866(10 119878) 1 5 9 85 853 17 173 3 3 17 173119866(10 119878) 1 2 8 9 75 953 15 193 4519 4519 15 193119866(10 119878) 1 2 5 8 9 65 35 13 7 137 137 13 7119866(10 119878) 1 2 3 7 8 9 60 752 12 152 85 85 12 152119866(10 119878) 1 2 3 5 7 8 9 55 40 11 8 118 118 11 8119866(10 119878) 1 2 3 4 6 7 8 9 50 852 10 172 2017 2017 10 172

Proof Obviously D and D119903 are irreducible for 119899 ge 4 ByLemma 1 we have

radicsum119899

119894=11198632

119894

119899le 120582 (119866) le max

1le119894le119899

119899

sum

119895=1

D119894119895radic119863119895

119863119894

radicsum119899

119894=1(119863119903)2

119894

119899le 120583 (119866) le max

1le119894le119899

119899

sum

119895=1

(D119903)119894119895

radic(119863119903)119895

(119863119903)119894

(8)

Note that 119866 is a regular graph of degree 119903 hence all the rowsums of D and D119903 are equal respectively that is 119863

1= 1198632=

sdot sdot sdot = 1198631198991198631199031= 119863119903

2= sdot sdot sdot = 119863

119903

119899 then we have

120582 (119866) = radicsum119899

119894=11198632

119894

119899 120583 (119866) =

radicsum119899

119894=1(119863119903)2

119894

119899

(9)

On the other hand according to definitions of the Wienerindex and the Harary index we have

119882(119866)

119867 (119866)=

sum119894lt119895

D119894119895

sum119894lt119895(D119903)119894119895

=

radicsum119899

119894=11198632

119894

radicsum119899

119894=1(119863119903)2

119894

(10)

Thus we obtain119882(119866)

119867 (119866)=120582 (119866)

120583 (119866) (11)

Theorem 6 Let 119866 be a r-circulant graph on 119899 vertices thenits Wiener index119882 and Harary index119867 listed below have thefollowing relationship

2119882

119899= 120582

2119867

119899= 120583 (12)

Proof Since a 119903-circulant graph is a 119903-regular graph applyingLemma 2 we get its maximum distance matrix eigenvalue

120582 = 2 (119899 minus 1) minus 119903 (13)

For 119903-regular graphs 119866 = (119881 119864) on 119899 vertices the followingequality holds

119899 sdot 119903 = 2 |119864| (14)

and we denote by |119864| the number of edges of 119866 According to(6) and (14) we have

119882 ge 119899 (119899 minus 1) minus119899119903

2 (15)

That is 2119882 ge 2119899(119899 minus 1) minus 119899119903 By (13) and (15) we have

2119882

119899ge 120582 (16)

On the other hand Indulal proved the following fact in [15]

Fact Let119866 be a graph withWiener index119882 Then 120582 ge 2119882119899and the equality holds if and only if 119866 is distance regular

Then from (16) and the previous fact we have 2119882119899 = 120582By equality (11) we obtain 2119867119899 = 120583 This completes the

proof of the theorem

4 Nonregular Graphs

As we just said for a regular those conclusions above holdBut for nonregular graph it is not easy to tell whether or notthe conclusions also hold

For arbitrary tree on 119899 vertices It is proved that (see [16])

(119899 minus 1)2le 119882(119879

119899) le (

119899 + 1

3) (17)

with equality on the left if and only if 119879119899cong 1198781119899minus1

and equalityon the right if and only if 119879

119899cong 119875119899 where 119875

119899is the path and

1198781119899minus1

is the star on 119899 verticesWithout loss of generality we now discuss the Wiener

index of graph 119870119896119899 it is obtained by joining 119896 pendent edges

to a vertex of complete graph 119870119899minus119896

It is well known that the Wiener index of 119866 has a direct

proportion with the distance spectral radius of 119866 as verticesfixed that is the distance spectral radius of 119866 is strictlyincreasing with the Wiener index increasing [15] Then weshow the following theorem

Theorem 7 Let 119866 be a simple connected graph on 119899 (119899 ge 4)vertices and 119896 cut edges then119882(119866) ge 119882(119870119896

119899)

Proof Wewill use the above well known fact to prove 120582(119866) ge120582(119870119896

119899) with equality if and only if 119866 cong 119870119896

119899

Let 1198641015840 = (1198901 1198902 119890

119896) be set of cut edges assume any

block of 119866 minus 1198641015840 has a clique If 119896 = 0 then 119866 cong 119870119896

119899 and the

proposition holdsNow suppose that 119896 ge 1 let 119870

1198860

1198701198861

119870119886119896

denote thecomponents of119866minus1198641015840 respectively where 119886

0+1198861+sdot sdot sdot+119886

119896= 119899

Let 119881119886119894

= V isin 119870119886119894

V is a endpoint of a cut edgeClaim 1 (|119881

119886119894

| = 1 0 le 119894 le 119896) In fact if 0 le 119894 le 119896 such that|119881119886119894

| gt 1 let 119906 V isin 119881119886119894

and VV119895 119906Vℎisin 1198641015840 Let 1198661015840 = 119866 minus VV

119895minus


119906Vℎ Obviously 1198661015840 is a disconnected graph on 119899 vertices and

119896 cut edges and dist1198661015840(Vℎ V119895) lt dist

119866(Vℎ V119895) Due to119873

119896119886119894

(119906)

V = 119873119896119886119894

(V) 119906 in 1198661015840 by Lemma 4 we have 120582(1198661015840) lt 120582(119866)and then we reach a contradiction

Without loss of generality assume 119881119886119894

= V119894 0 le 119894 le 119896

Claim 2 If V119904is adjacent to V

119905 0 le 119904 119905 le 119896 then 119889(V

119904) = 1 or

119889(V119905) = 1

In fact if 119889(V119904) ge 2 and 119889(V

119905) ge 2

Let1198621 1198622denote two components containing V

119904 V119905in119866minus

V119904V119905 respectively and 119906 isin 119873(V

119904) V119905 = V

119896+1 V119896+2 V

119896+119903

Let1198661015840 = 119866minusV119904V119896+1minussdot sdot sdotminusV

119904V119896+119903+V119905V119896+1+sdot sdot sdot+V

119905V119896+119903

Then1198661015840 still is a connected graph with 119899 vertices and 119896 cut edges

and dist1198661015840(119906 V119896+119894) lt dist

119866(119906 V119896+119894) (1 le 119894 le 119903) where V

119904V119905is a

pendent edge in 1198661015840 Using Lemma 4 we have 120582(1198661015840) lt 120582(119866)and this leads to a contradiction

Since 119866 is connected according to Claims 1 and 2 weobtain 119866 cong 119870119896

119899 Hence we have119882(119866) ge 119882(119870119896

119899)

This completes the proof of the theorem

Conflict of Interests

The author declares that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The author is grateful to the reviewers for their valuable com-ments and suggestions Project is supported by Hunan Pro-vincial Natural Science Foundation of China no 13JJ3118

References

[1] F K Hwang ldquoA survey on multi-loop networksrdquo TheoreticalComputer Science vol 299 no 1ndash3 pp 107ndash121 2003

[2] D Janezic AMilicevic SNikolic andNTrinajsticGraph-The-oretical Matrices in Chemistry Mathematical Chemistry Mono-graphs No 3 University of Kragujevac Kragujevac Serbia2007

[3] O Ivanciuc T Ivanciuc and A T Balaban ldquoQuantitativestructure-property relationship evaluation of structural desc-riptors derived from the distance and reverseWiener matricesrdquoInternet Electronic Journal of Molecular Design vol 1 pp 467ndash487 2002

[4] H Hosoya ldquoTopological indexrdquo Bulletin of the Chemical Societyof Japan vol 44 no 9 pp 2332ndash2339 1971

[5] J Devillers and A T Balaban Topological Indices and RelatedDescriptors in QSAR and QSPR Gordon and Breach Amster-dam The Netherlands 1999

[6] T S Balaban P A Filip andO Ivanciuc ldquoComputer generationof acyclic graphs based on local vertex invariants and topologi-cal indices Derived canonical labelling and coding of trees andalkanesrdquo Journal of Mathematical Chemistry vol 11 no 1 pp79ndash105 1992

[7] O Ivanciuc T Ivanciuc and A T Balaban ldquoDesign of topologi-cal indices Part 101 Parameters based on electronegativity andcovalent radius for the computation ofmolecular graph descrip-tors for heteroatom-containing moleculesrdquo Journal of Chemical

Information and Computer Sciences vol 38 no 3 pp 395ndash4011998

[8] M V Diudea ldquoIndices of reciprocal properties or Harary indi-cesrdquo Journal of Chemical Information and Computer Sciencesvol 37 no 2 pp 292ndash299 1997

[9] R Todeschini andVConsonniHandbook ofMolecularDescrip-tors Wiley-VCH Weinheim Germany 2000

[10] L Hogben ldquoSpectral graph theory and the inverse eigenvalueproblem of a graphrdquo Electronic Journal of Linear Algebra vol14 pp 12ndash31 2005

[11] Z Bo ldquoOn the Spectral radius of nonnegative matricesrdquo Aus-tralasian Journal of Combinatorics vol 22 pp 301ndash306 2000

[12] G Indulal I Gutman andAVijayakumar ldquoOn distance energyof graphsrdquoMatch vol 60 no 2 pp 461ndash472 2008

[13] L Xiaoxia ldquoOn the extremal Wiener indices of some graphsrdquoOR Transactions vol 14 no 2 pp 55ndash60 2010

[14] S N Ruzieh ldquoThe distance spectrum of the path 119875119899and the first

distance eigenvector of conneeted graphrdquo Linear Algebra and ItsApplications vol 281 pp 75ndash81 1990

[15] G Indulal ldquoSharp bounds on the distance spectral radiusand the distance energy of graphsrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 106ndash113 2009

[16] R C Entringer D E Jackson and D A Snyder ldquoDistance ingraphsrdquo Czechoslovak Mathematical Journal vol 26 pp 283ndash296 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


Balaban et al [6] The maximum eigenvalues of variousmatrices have recently attracted attention of mathematicalchemists [6ndash8]

The Harary index and the related indices such as itsextension to heterosystems [7] and the hyper-Harary index[8] have shown a modest success in structure-property cor-relations [9] but the use of these indices in combination withother descriptors appears to be very efficacious in improv-ing the QSPR (quantitative structure-property relationship)models

The paper is organized as follows In Section 2 we bringforward an interesting phenomenon In Section 3 we analyzethe cause of this phenomenon to emerge we report ourresults for the maximum eigenvalues of the Wiener matrixand the Harary matrix of 119903-circulant graphs Finally inSection 4 we discuss theWiener index of nonregular graphs

2 An Interesting Phenomenon

Recall that for a positive integer 119899 and set 119878 sube 0 1 2 119899 minus1 the circulant graph 119866(119899 119878) is the graph with 119899 verticeslabeled with integers modulo 119899 such that each vertex 119894 isadjacent to 119878 other vertices 119894 + 119904 (mod 119899) | 119904 isin 119878 Theset 119878 is called a symbol of 119866(119899 119878) As we will consider onlyundirected graphs without loops we assume that 0 notin 119878 and119904 isin 119878 if and only if 119899 minus 119904 isin 119878 and therefore the vertex 119894 isadjacent to vertices 119894 plusmn 119904 (mod 119899) for each 119904 isin 119878 In otherwords a graph is circulant if it is Cayley graph on the circulantgroup that is its adjacency matrix is circulant

For any circulant graph 119866 it is a regular graph A 119903-circulant graph is called regular of degree (or valency) 119903when every vertex has precisely 119903 neighbors Let 119889 denotethe diameter of 119866 then 119889 is less than the number of distincteigenvalues of the adjacency matrix of 119866 (see [10])

Now consider 4-circulant graphs with 6 7 18 ver-tices respectively By a straightforwardmathematical calcula-tion we obtain somedata on theWiener index119882 theHararyindex119867 and the eigenvalues 120582 120583 of the Wiener matrix andthe Harary matrix as follows in Table 1

From Table 1 it is easy to observe that119882119867 and 120582120583 areequal Moreover we have 2119882119899 = 120582 and 2119867119899 = 120583

We again observe Table 2From Table 2 we also obtain the same results previously

when the number of vertices is fixed in 119866(119899 119878)Why does this phenomenon occur We will be interested

in trying to explain something about this phenomenon Toprove our result we need a few more lemmas

Lemma 1 (see [11]) Let B = (B119894119895) be an 119899 times 119899 nonnegative

irreducible symmetric matrix (119899 ge 2) with row sums1198611 1198612 119861

119899 If 120588(B) is the maximum eigenvalue of B then

radicsum119899

119894=11198612

119894

119899le 120588 (B) le max

1le119894le119899

119899

sum

119895=1

B119894119895radic119861119895

119861119894

(4)

Table 1

119899 119882 119867 119889 120582 120583 119882119867 120582120583 2119882119899 2119867119899

6 18 135 2 6 45 43 43 6 457 28 175 2 8 5 85 85 8 58 40 22 2 10 55 2011 2011 10 559 54 27 2 12 6 2 2 12 610 75 953 3 15 193 4519 4519 15 19311 99 1103 3 18 203 2710 2710 18 20312 126 42 3 21 7 3 3 21 713 156 1433 3 24 223 3611 3611 24 22314 196 63712 4 28 9112 4813 4813 28 911215 240 2354 4 32 476 19247 19247 32 47616 288 1943 4 36 9712 43297 43297 36 971217 340 4256 4 40 253 245 245 40 25318 405 3845 5 45 12815 675128 675128 45 12815

with either equality if and only if 1198611= 1198612= sdot sdot sdot = 119861

119899or there

is a permutation matrix Q such that

Q119879BQ = (0 CC119879 0) (5)

where all the row sums of C are equal

Lemma 2 (see [12]) Let 119866 be a r-regular graph of order 119899If 119903 120582

2 120582

119899are the adjacency eigenvalues of 119866 then its D-

eigenvalues are 2119899minus119903minus2 andminus(120582119894+2) 119894 = 2 3 119899 where D-

eigenvalues denote the eigenvalues of the distancematrix119863(119866)

Lemma 3 (see [13]) Let 119866 be a r-circulant graph on 119899 verticesand the Wiener index 119882 of 119866 then the following equalityholds

119882 ge 119899 sdot (119899 minus 1) minus |119864| (6)

where |119864| denotes the number of edges

Lemma 4 (see [14]) Let 1198621be a component of 119866 minus 119906 and

V1 V2 V

119896(1 le 119896 le deg

119866(119906)minusdeg

1198621

(119906)) isin 119873119866(119906)119873

1198621

(119906)If 1198731198621

(119906) V = 1198731198621

(V) where V is adjacent to 119906 in 1198621

let 1198661015840 be obtained by removing an edge 119906V119904of 119866 and add a

new edge VV119904(1 le 119904 le 119896) For all 1 le 119904 le 119896 if a vertex

119908 isin 119881(119866) 119881(1198621) cup 119906 satisfies dist

119866(119908 V119904) lt dist

1198661015840(119908 V119904)

then 120582(119866) lt 120582(1198661015840) where dist

119866(119906 V) denotes the distance

between a pair of vertices 119906 V in 119866

3 Main Results

In this section we will prove the following two theorems

Theorem 5 Let119866 be a r-circulant graph on 119899 (119899 ge 4) verticesthen the following equality holds

119882(119866)

119867 (119866)=120582 (119866)

120583 (119866) (7)


Table 2

Graph 119878 119882 119867 120582 120583 119882119867 120582120583 2119882119899 2119867119899

119866(10 119878) 1 5 9 85 853 17 173 3 3 17 173119866(10 119878) 1 2 8 9 75 953 15 193 4519 4519 15 193119866(10 119878) 1 2 5 8 9 65 35 13 7 137 137 13 7119866(10 119878) 1 2 3 7 8 9 60 752 12 152 85 85 12 152119866(10 119878) 1 2 3 5 7 8 9 55 40 11 8 118 118 11 8119866(10 119878) 1 2 3 4 6 7 8 9 50 852 10 172 2017 2017 10 172


radicsum119899

119894=11198632

119894

119899le 120582 (119866) le max

1le119894le119899

119899

sum

119895=1

D119894119895radic119863119895

119863119894

radicsum119899

119894=1(119863119903)2

119894

119899le 120583 (119866) le max

1le119894le119899

119899

sum

119895=1

(D119903)119894119895

radic(119863119903)119895

(119863119903)119894

(8)


1= 1198632=

sdot sdot sdot = 1198631198991198631199031= 119863119903


119903

119899 then we have

120582 (119866) = radicsum119899

119894=11198632

119894

119899 120583 (119866) =

radicsum119899

119894=1(119863119903)2

119894

119899

(9)


119882(119866)

119867 (119866)=

sum119894lt119895

D119894119895

sum119894lt119895(D119903)119894119895

=

radicsum119899

119894=11198632

119894

radicsum119899

119894=1(119863119903)2

119894

(10)


119867 (119866)=120582 (119866)

120583 (119866) (11)


2119882

119899= 120582

2119867

119899= 120583 (12)


120582 = 2 (119899 minus 1) minus 119903 (13)


119899 sdot 119903 = 2 |119864| (14)


119882 ge 119899 (119899 minus 1) minus119899119903

2 (15)


2119882

119899ge 120582 (16)





4 Nonregular Graphs



(119899 minus 1)2le 119882(119879

119899) le (

119899 + 1

3) (17)



119899cong 119875119899 where 119875


1198781119899minus1







119899)



119899

Let 1198641015840 = (1198901 1198902 119890



119899 and the


1198860

1198701198861

119870119886119896


0+1198861+sdot sdot sdot+119886

119896= 119899

Let 119881119886119894

= V isin 119870119886119894


119886119894


| gt 1 let 119906 V isin 119881119886119894


119895minus




119866(Vℎ V119895) Due to119873

119896119886119894

(119906)

V = 119873119896119886119894



= V119894 0 le 119894 le 119896


119905 0 le 119904 119905 le 119896 then 119889(V

119904) = 1 or

119889(V119905) = 1


119905) ge 2


119904 V119905in119866minus


119904) V119905 = V

119896+1 V119896+2 V

119896+119903



119905V119896+119903


and dist1198661015840(119906 V119896+119894) lt dist

119866(119906 V119896+119894) (1 le 119894 le 119903) where V

119904V119905is a



119899 Hence we have119882(119866) ge 119882(119870119896

119899)




Acknowledgments


References




























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Table 2

Graph 119878 119882 119867 120582 120583 119882119867 120582120583 2119882119899 2119867119899

119866(10 119878) 1 5 9 85 853 17 173 3 3 17 173119866(10 119878) 1 2 8 9 75 953 15 193 4519 4519 15 193119866(10 119878) 1 2 5 8 9 65 35 13 7 137 137 13 7119866(10 119878) 1 2 3 7 8 9 60 752 12 152 85 85 12 152119866(10 119878) 1 2 3 5 7 8 9 55 40 11 8 118 118 11 8119866(10 119878) 1 2 3 4 6 7 8 9 50 852 10 172 2017 2017 10 172


radicsum119899

119894=11198632

119894

119899le 120582 (119866) le max

1le119894le119899

119899

sum

119895=1

D119894119895radic119863119895

119863119894

radicsum119899

119894=1(119863119903)2

119894

119899le 120583 (119866) le max

1le119894le119899

119899

sum

119895=1

(D119903)119894119895

radic(119863119903)119895

(119863119903)119894

(8)


1= 1198632=

sdot sdot sdot = 1198631198991198631199031= 119863119903


119903

119899 then we have

120582 (119866) = radicsum119899

119894=11198632

119894

119899 120583 (119866) =

radicsum119899

119894=1(119863119903)2

119894

119899

(9)


119882(119866)

119867 (119866)=

sum119894lt119895

D119894119895

sum119894lt119895(D119903)119894119895

=

radicsum119899

119894=11198632

119894

radicsum119899

119894=1(119863119903)2

119894

(10)


119867 (119866)=120582 (119866)

120583 (119866) (11)


2119882

119899= 120582

2119867

119899= 120583 (12)


120582 = 2 (119899 minus 1) minus 119903 (13)


119899 sdot 119903 = 2 |119864| (14)


119882 ge 119899 (119899 minus 1) minus119899119903

2 (15)


2119882

119899ge 120582 (16)





4 Nonregular Graphs



(119899 minus 1)2le 119882(119879

119899) le (

119899 + 1

3) (17)



119899cong 119875119899 where 119875


1198781119899minus1







119899)



119899

Let 1198641015840 = (1198901 1198902 119890



119899 and the


1198860

1198701198861

119870119886119896


0+1198861+sdot sdot sdot+119886

119896= 119899

Let 119881119886119894

= V isin 119870119886119894


119886119894


| gt 1 let 119906 V isin 119881119886119894


119895minus




119866(Vℎ V119895) Due to119873

119896119886119894

(119906)

V = 119873119896119886119894



= V119894 0 le 119894 le 119896


119905 0 le 119904 119905 le 119896 then 119889(V

119904) = 1 or

119889(V119905) = 1


119905) ge 2


119904 V119905in119866minus


119904) V119905 = V

119896+1 V119896+2 V

119896+119903



119905V119896+119903


and dist1198661015840(119906 V119896+119894) lt dist

119866(119906 V119896+119894) (1 le 119894 le 119903) where V

119904V119905is a



119899 Hence we have119882(119866) ge 119882(119870119896

119899)




Acknowledgments


References




























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119866(Vℎ V119895) Due to119873

119896119886119894

(119906)

V = 119873119896119886119894



= V119894 0 le 119894 le 119896


119905 0 le 119904 119905 le 119896 then 119889(V

119904) = 1 or

119889(V119905) = 1


119905) ge 2


119904 V119905in119866minus


119904) V119905 = V

119896+1 V119896+2 V

119896+119903



119905V119896+119903


and dist1198661015840(119906 V119896+119894) lt dist

119866(119906 V119896+119894) (1 le 119894 le 119903) where V

119904V119905is a



119899 Hence we have119882(119866) ge 119882(119870119896

119899)




Acknowledgments


References




























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

Date post:	01-Jun-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

Research Article The Wiener Index of Circulant...

Documents