On Super Mean Labeling for Line and Middle Graph
of Path and Cycle
1. D.Tresa Roselin
2. Dr.Sr.Stanis Arul Mary
1. PG Scholar, Department of Mathematics, Nirmala College for Women,
Red fields, Coimbatore.
E-mail: [email protected]
2. Assistant Professor, Department of Mathematics, Nirmala College for women,
Red fields, Coimbatore.
Abstract:
Let G be a (π, π) graph. A (π, π) graph is a graph with π vertices and π edges. Let us define
an injection mapping π βΆ π(πΊ) β {1,2,β¦ , π + π} , for each edge π = π’π£ in E labeled by,
πβ(π) = {
π(π’) + π(π£)
2 ππ π(π’) + π(π£) ππ ππ£ππ
π(π’) + π(π£) + 1
2 ππ π(π’) + π(π£)ππ πππ
Then f is said to be super mean labeling if the set
π(π£) βͺ {πβ(π): πππΈ} = {1,2,β¦ , π + π}.
The graph which admits super mean labeling is called super mean graph.
In this paper we have found out the Super Mean Labeling of Line and middle Graph of Path
and Cycle.
Keywords:
Super Mean Labeling, Line graph of path with n vertices - πΏ(ππ), Line Graph of cycle with n
vertices - πΏ(πΆπ), Middle Graph of path with n vertices - π(ππ),Middle Graph of cycle with n
vertices - π(πΆπ).
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Introduction:
A graph G is a combination of vertices and edges which are connected to each other.Let G (V,E)
be a graph with vertex set V and the edge set E, respectively. By a graphG = (V,E) we mean a
finite undirected graph with neither loops nor multiple edges. In this paper π(πΊ) is the vertex set
and πΈ(πΊ) is the edge set of a graph G.
The number of vertices of G is called order of G and it is denoted by π. The number of edges of
G is called size of G and it is denoted by π. A(π, π)graph G is a graph with π vertices and π
edges.Terms and notations no defined here are used in the sense of Harary[1].
A graph labeling is the assignment of labels traditionally represented by integers,to vertices
and/or edges of a graph. We have different methods of labeling.
Graph labeling was first introduced in the late 1960βs. Many studies in graph labeling was made
which refers to Rosaβs research and that was held in 1967[6].
In 2003, Somasundaram and Ponraj [2] have introduced the notation of Mean Labelingof graphs.
The concept of Super Mean Labeling was introduced by Ponraj and Ramya [3].Furthermore
some results on super mean graphs are discussed in[7-11].
B. Gayathri and M. Tamilselvi[12,13] extended super mean labeling to k-super mean labeling.
In 2018, NurInayah, I.WayanSudarsanaSelvyMusdalifah, and Nurhasanah Daeng Mangesa have
introduced the notion of Super Mean Labeling for Total Graph of Path and Cycle[4].
Preliminaries:
Definition 1:
Let G be a (π, π) graph. A graph G is called a mean graph if there is an injective function π
from the vertices of G to {0,1,2,β¦ , π}. Such that when each edge π = π’π£ is labeled with
πβ(π) = {
π(π’) + π(π£)
2 ππ π(π’) + π(π£)is even
π(π’) + π(π£) + 1
2 ππ π(π’) + π(π£)is odd
Then the resulting edge labels are distinct.
Definition 2:
Let π: π β {1,2,β¦ . π + π} be an injection on G. For each edge π = π’π£ and an integer π β₯2, the induced Smarandachely edge m-labeling πβis defined by
πβ(π = ππ) =π(π)+π(π)
π.
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Then πis called a Smarandachely super m-mean labeling if
π(π£) βͺ {πβ(π): πππΈ} = {1,2,β¦ , π + π}.
A graph that admits super mean m-labeling is called Smarandachely m-mean graph.
Particularly, ifπ = π, we know that,
πβ(π = π’π£) = {
π(π’) + π(π£)
2 ππ π(π’) + π(π£)ππ ππ£ππ
π(π’) + π(π£) + 1
2 ππ π(π’) + π(π£)ππ πππ
Such a labeling π is called super mean labeling of G if
π(π) βͺ {πβ(π): π π πΈ} = {1,2, . . , π + π}.
A graph that admits super mean labeling is called super mean graph.
Definition 3:
The Line Graph L(G) of G is the graph with vertices are the edges of G with two vertices of
L(G) adjacent whenever the corresponding edges of G are adjacent[5].
Definition 4:
The Middle Graph M(G) is defined as follows[5]. The vertex set M(G) is V(G)βͺE(G).
Twovertices X,Y in the vertex set M(G)are adjacent in M(G) in case one of the following holds:
i. X , Y are in E(G) and X,Y are adjacent in G .
ii. X is in V(G), Y is in E(G), and X,Y are incident in G.
On Super Mean Labeling for Line Graph of Path:
The theorem proposed in this section deals with the super mean labeling for line graph of path
on n vertices, πΏ(ππ).
Theorem 1:
The line graph of path on n vertices, L(Pn) is a super mean graph for all n β₯ 3.
Proof:
Let π(πΏ(ππ)) = {π’π: 1 β€ π β€ π β 1} and πΈ(πΏ(ππ)) = {ππ, 1 β€ π β€ π β 2} with ei = uiui+1
for 1 β€ π β€ π β 2. Immediately, we have that the cardinality of the vertex set and the edge set of
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πΏ(ππ)areπ = π β 1 and π = π β 2respectively, and so π + π = 2π β 3. Define an injection
π: π(πΏ(ππ)) β {12,β¦ 2π β 2}for π β₯ 3 as follows:
π(π’π) = 2π β 1 πππ π = 1,2,β¦ π β 1
And so we have,
πβ(ππ) = 2π πππ π = 1,2,β¦ , π β 2
Next, we consider the following sets:
π΄1 = {π(π’π) = 2π β 1 πππ π = 1,2,β¦ π β 1}
A2 = {πβ(ππ) = 2π πππ π = 1,2, β¦ , π β 2}
It can be verified that π (π(πΏ(ππ))) βͺ πβ (πΈ(πΏ(ππ))) = β π΄π = {1,2, β¦ , π β 1}
2π=1 and so π is
a super mean labeling of πΏ(ππ). Hence πΏ(ππ) is a super mean graph.
On Super Mean Labeling for Line Graph of Cycle:
The theorem proposed in this section deals with the super mean labeling for line graph of cycle
of n vertices, πΏ(πΆπ).
Theorem 2:
The line graph of cycle on n vertices,πΏ(πΆπ), is a super mean graph if either π is odd and π β₯ 3
or π is even and π β₯ 6.
Proof:
Let π(πΏ(πΆπ)) = {π’π: 1 β€ π β€ π}and πΈ(πΏ(πΆπ)) = {ππ, 1 β€ π β€ π}with
ππ = { π’ππ’π+1 πππ 1 β€ π β€ π β 1π’ππ’1 πππ π = π
Immediately, we have that the cardinality of the vertex set and the edge set of πΏ(πΆπ) areπ =π πππ π = π respectively, and so π + π = 2π.
Define an injection π: π(πΏ(πΆπ)) β {1,2,β¦ ,2π} for odd andπ β₯ 3 as follows.
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π(π’π) = {2π β 1 πππ π = 1,2,β¦
π + 1
2
2π πππ π =π + 1
2+ 1, β¦ , π
And also we have,
πβ(ππ) =
{
2π πππ π = 1,2,β¦ ,
π β 1
2
2π + 1 πππ π =π + 1
2,β¦π β 1
π + 1 πππ π = π
Next, we consider the following sets,
π΄1 = {π(π’π) = 2π β 1 πππ π = 1,2,β¦ ,π + 1
2}
π΄2 = {π(π’π) = 2π πππ π =π + 1
2+ 1,β¦ , π}
π΄3 = {πβ(ππ) = 2π πππ π = 1,2, β¦ ,
π β 1
2}
π΄4 = {πβ(ππ) = 2π + 1 πππ π =
π + 1
2, β¦π β 1}
π΄5 = {πβ(ππ) = π + 1 πππ π = π}
It can be verified that π(π(πΏ(πΆπ))) βͺ πβ(πΈ(πΏ(πΆπ))) = β π΄π = {1,2,β¦ 2π}
5π=1 and so f is a super
mean labeling ofπΏ(πΆπ). Hence πΏ(πΆπ) is a super mean labeling for line graph of cycle when π is
odd and π β₯ 3.
Now define an injection π1: π(πΏ(πΆπ)) β {1,2, β¦ 2π} for π is even and π β₯ 6 as follows
π1(π’π) =
{
2π β 1 πππ π = 1,2 3π β 3 πππ π = 3,4
2π + 1 πππ π = 5,β¦ ,π
2+ 1
2π + 2 πππ π
2+ 2,β¦ , π β 1
7 πππ π = π
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And also we have,
π1β(ππ) =
{
2 πππ π = 1 3π β 1 πππ π = 2,3
2π + 2 πππ π = 4,β¦π
2
2π + 3 πππ π =π
2+ 1,β¦ , π β 2
π + 4 πππ π = π β 1 4 πππ π = π
Next we consider the following sets,
π΄1 = {π1(π’π) = 2π β 1 πππ π = 1,2}
π΄2 = {π1(π’π) = 3π β 3 πππ π = 3,4}
π΄3 = {π1(π’π) = 2π + 1 πππ π = 5,β¦ ,π
2+ 1}
π΄4 = {π1(π’π) = 2π + 2 πππ π =π
2+ 2,β¦ , π β 1}
π΄5 = {π1(π’π) = 7 πππ π = π}
π΄6 = {π1β(ππ) = 2 πππ π = 1}
π΄7 = {π1β(ππ) = 3π β 1 πππ π = 2,3}
π΄8 = {π1β(ππ) = 2π + 2 πππ π = 4,β¦ .
π
2}
π΄9 = {π1β(ππ) = 2π + 3 πππ π =
π
2+ 1, β¦ , π β 2}
π΄10 = {π1β(ππ) = π + 4 ππππ = π β 1}
π΄11 = {π1β(ππ) = 4 πππ π = π}
It can be verified that π1(π(πΏ(πΆπ))) βͺ π1β(πΈ(πΏ(πΆπ))) = β π΄π = {1,2, β¦ 2π}
11π=1 and so π1 is a
super mean labeling of πΏ(πΆπ). Hence πΏ(πΆπ) is a super mean labeling for line graph of cycle
when π is evenand π β₯ 6.
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On Super Mean Labeling for Middle Graph of Path:
The theorem proposed in this section deals with the super mean labeling for middle graph of
path on π vertices, π(ππ).
Theorem 3:
The middle graph of path on π vertices, π(ππ), is a super mean graph for all π β₯ 3.
Proof:
Let π(π(ππ)) = {π£π: 1 β€ π β€ π} βͺ {π’π: 1 β€ π β€ π β 1} and πΈ(π(ππ)) = {ππΚΉ, ππ
ΚΊ: 1 β€ π β€
π β 1} βͺ {ππ: 1 β€ π β€ π β 2} with ei = uiui+1 for 1 β€ π β€ π β 2, ππΚΉ = π£ππ’π and ππ
ΚΊ =
π’ππ£π+1 πππ 1 β€ π β€ π β 1
Immediately, we have that the cardinality of the vertex set and the edge set of π(ππ) are π =2π β 1 and π = 3π β 4 respectively, and so π + π = 5π β 5.
Define an injection π: π(π(ππ)) β {1,2,β¦ 5π β 5}for π β₯ 3 as follows:
π(π£π) = {1 πππ π = 1
5π β 5 πππ π = 2,β¦π
Also, π(π’π) = 5π β 2 πππ 1 β€ π β€ π β 1
And so we have,
πβ(ππ) = 5π + 1 πππ π = 1,2,β¦ , π β 2
πβ(ππΚΉ) = 5π β 3 πππ π = 1,2,β¦ , π β 1
πβ(ππΚΊ) = 5π β 1 πππ π = 1,2,β¦ , π β 1
Next, we consider the following sets:
π΄1 = {π(π£π) = 1 πππ π = 1}
π΄2 = {π(π£π) = 5π β 5 πππ π = 2, β¦ , π}
π΄3 = {π(π’π) = 5π β 2 πππ π = 1,2,β¦ π β 1}
A4 = {πβ(ππ) = 5π + 1 πππ π = 1,2,β¦ , π β 2}
A5 = {πβ(ππ
ΚΉ) = 5π β 3 πππ π = 1,2, β¦ , π β 1}
A6 = {πβ(ππ
ΚΊ) = 5π β 1 πππ π = 1,2,β¦ , π β 1}
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It can be verified that π (π(π(ππ))) βͺ πβ (πΈ(π(ππ))) = β π΄π = {1,2,β¦ ,5π β 5}
6π=1 and so π
is a super mean labeling of π(ππ). Hence π(ππ) is a super mean graph.
On Super Mean Labeling for Middle Graph of Cycle:
The theorem proposed in this section deals with the super mean labeling for middle graph of
cycle of n vertices, π(πΆπ).
Theorem 4:
The middle graph of cycle on n vertices, π(πΆπ), is a super mean graph if either π is odd and
π β₯ 3 or π is even and π β₯ 4.
Proof:
Let π(π(πΆπ)) = {π£π ,π’π: 1 β€ π β€ π} and πΈ(π(πΆπ)) = {ππ, ππΚΉ, ππ
ΚΊ: 1 β€ π β€ π} with
ππ = { π’ππ’π+1 πππ 1 β€ π β€ π β 1π’ππ’1 πππ π = π
ππΚΉ = π’ππ£π πππ π = 1,2,β¦ π
ππΚΊ = {
π£ππ’π+1 πππ 1 β€ π β€ π β 1π£ππ’1 πππ π = π
Immediately, we have that the cardinality of the vertex set and the edge set of π(πΆπ) are
π = 2π πππ π = 3πrespectively, and soπ + π = 5π.
Define an injection π: π(π(πΆπ)) β {1,2,β¦ ,5π} for odd andπ β₯ 3 as follows.
π(π£π) =
{
5π β 2 πππ π = 1,2,β¦
πβ1
2
5π + 1 πππ π =π+1
2
5π πππ π =π+3
2, β¦ , π
π(π’π) =
{
1 πππ π = 1
5π β 4 πππ π = 2,3, β¦ ,π + 1
2
5π β 2 πππ π =π + 3
2, β¦ , π
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And also we have,
πβ(ππ) =
{
4 πππ π = 1
5π β 1 πππ π = 2,3,β¦π β 1
2
5π πππ π =π + 1
2
5π + 1 πππ π =π + 3
2, β¦ , π β 1
5π β 1
2 πππ π = π
πβ(ππΚΉ) =
{
2 πππ π = 1
5π β 3 πππ π = 2,3,β¦ ,π β 1
2
5π β 1 πππ π =π + 1
2, β¦ , π
πβ(ππΚΊ) =
{
5π πππ π = 1,2,β¦ ,
π β 1
2
5π β 2 πππ π =π + 1
2, β¦ , π β 1
5π + 1
2 πππ π = π
Next, we consider the following sets,
π΄1 = {π(π£π) = 5π β 2 πππ π = 1,2, β¦ ,π β 1
2}
π΄2 = {π(π£π) = 5π + 1 πππ π =π+1
2}
π΄3 = {π(π£π) = 5π πππ π =π + 3
2, β¦ , π}
π΄4 = {π(π’π) = 1 πππ π = 1}
π΄5 = {π(π’) = 5π β 4 πππ π = 2,3,β¦ ,π + 1
2}
π΄6 = {π(π’π) = 5π β 2 πππ π =π + 3
2, β¦ , π}
π΄7 = {πβ(ππ) = 4 πππ π = 1}
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π΄8 = {πβ(ππ) = 5π β 1 πππ π = 2,3,β¦ ,
π β 1
2}
π΄9 = {πβ(ππ) = 5π πππ π =
π + 1
2}
π΄10 = {πβ(ππ) = 5π + 1 πππ π =
π + 3
2, β¦ , π β 1}
π΄11 = {πβ(ππ) =5π β 1
2 πππ π = π}
π΄12 = {πβ(ππ
ΚΉ) = 2 πππ π = 1}
π΄13 = {πβ(ππ
ΚΉ) = 5π β 3 πππ π = 2,3, β¦ ,π β 1
2}
π΄14 = {πβ(ππΚΉ) = 5π β 1 πππ π =
π + 1
2, β¦ , π}
π΄15 = {πβ(ππΚΊ) = 5π πππ π = 1,2,β¦ ,
π β 1
2}
π΄16 = {πβ(ππ
ΚΊ) = 5π + 2 πππ π =π + 1
2,β¦ , π β 1}
π΄17 = {πβ(ππ
ΚΊ) =5π + 1
2 πππ π = π}
It can be verified that π(π(π(πΆπ))) βͺ πβ(πΈ(π(πΆπ))) = β π΄π = {1,2,β¦ 5π}
17π=1 and sof is a
super mean labeling of π(πΆπ). Hence π(πΆπ) is a super mean labeling for line graph of cycle
when π is odd andπ β₯ 3.
Now define an injection π1: π(π(πΆπ)) β {1,2,β¦ 5π} for π is even and π β₯ 6 as follows,
π1(π£π) =
{
1 ππππ = 1
5π β 5 ππππ = 2,3,β¦ π
2β 1
5π β 2 ππππ =π
2 , β¦ ,
π
2+ 1
5π β 3 ππππ
2+ 2,β¦ , π
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π1(π’π) =
{
5π β 2 πππ π = 1,2,β¦ ,
π
2β 1
5π β 5 πππ π =π
2
5π πππ π
2+ 1, β¦ , π
And also we have,
π1β(ππ) =
{
5π + 1 πππ π = 1 ,2,β¦
π
2β 2
5π β 1 πππ π =π
2β 1
5π πππ π =π
2
5π + 3 πππ π =π
2+ 1,β¦ , π β 1
5π + 4
2 πππ π = π
π1β(ππ
ΚΉ) =
{
5π β 1
2 πππ π = 1
5π β 3 πππ π = 2,β¦ ,π
2
5π β 1 πππ π =π
2+ 1, β¦ , π
π1β(ππ
ΚΊ) =
{
5π β 1 πππ π = 1,β¦ ,
π
2β 2 πππ
π
2
5π + 1 πππ π =π
2β 1 πππ
π
2+ 1, β¦ , π β 1
5π + 2
2 πππ π = π
Next we consider the following sets,
π΄1 = {π1(π£π) = 1 πππ π = 1}
π΄2 = {π1(π£π) = 5π β 5 πππ π = 2,3,β¦ ,π
2β 1}
π΄3 = {π1(π£π) = 5π + 2 πππ π =π
2,β¦ ,
π
2+ 1}
π΄4 = {π1(π£π) = 5π β 3 πππ π =π
2+ 2,β¦ , π}
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π΄5 = {π1(π’π) = 5π β 2 πππ π = 1,2,β¦ ,π
2β 1}
π΄6 = {π1(π’π) = 5π β 5 πππ π =π
2}
π΄7 = {π1(π’π) = 5π πππ π =π
2+ 1, β¦ , π}
π΄8 = {π1β(ππ) = 5π + 1 πππ π = 1,2, β¦
π
2β 2}
π΄9 = {π1β(ππ) = 5π β 1 πππ π =
π
2β 1}
π΄10 = {π1β(ππ) = 5π πππ π =
π
2}
π΄11 = {π1β(ππ) = 5π + 3 πππ π =
π
2+ 1,β¦ , π β 1}
π΄12 = {π1β(ππ) =
5π + 4
2ππππ = π}
π΄13 = {π1β(ππ
ΚΉ) =5π β 1
2 πππ π = 1}
π΄14 = {π1β(ππ
ΚΉ) = 5π β 3 πππ π = 2,β¦ ,π
2}
π΄15 = {π1β(ππ
ΚΉ) = 5π β 1 πππ π =π
2+ 1,β¦ , π}
π΄16 = {π1β(ππ
ΚΊ) = 5π β 1 πππ π = 1,β¦π
2β 2 πππ
π
2 }
π΄17 = {π1β(ππ
ΚΉ) = 5π + 1 πππ π =π
2β 1 πππ
π
2+ 1, β¦ , π β 1}
π΄18 = {π1β(ππ
ΚΉ) =5π + 2
2 πππ π = π}
It can be verified that π1(π(π(πΆπ))) βͺ π1β(πΈ(π(πΆπ))) = β π΄π = {1,2,β¦ 5π}
18π=1 and so π1 is
a super mean labeling ofπ(πΆπ). Hence π(πΆπ) is a super mean labeling for line graph of cycle
when π is even and π β₯ 6.
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Summary and Remarks:
Here we propose new results corresponding to super mean labeling for line and middle graph
of path and cycle. All results reported here are in line and middle graph of path and
cycle,πΏ(ππ), πΏ(πΆπ), π(ππ)and π(πΆπ). In future, it is not only possible to investigate some
more results corresponding to other graph families but also Smarandachely super m-mean
labeling in general as well.
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