EDGE MEAN LABELING OF A REGULAR GRAPHS K.SuriyaP rabha1
Department of Mathematics Alagappa University, karaikudi-630 003
Tamilnadu, India.
Email: suriyaprabha23@gmail.com S.Amutha2
Ramaujan Centre for Higher Mathematics(RCHM) Alagappa University, karaikudi-630 003
Tamilnadu, India.
Email: amutha.angappan@rediffmail.com
Abstract
In this paper, we introduce a concept of edge odd and even mean labeling of a regular graph and also investigate the behaviour of some standard graphs.
Key words: Mean labeling, edge odd mean labeling, edge even mean labeling.
AMS Subject Classification :05c
1
Introduction
As a standard notation, assume thatG=G(V, E) is a finite, simple and undirected graph with p vertices and q edges. Terms and terminology as in Harary [3]. Mean labeling was introduced by S.Somasundaram and R.Ponraj in [6]. In this paper, we study the edge odd and even mean labeling of a regular graph obtained by joining some standard graphs.
Path on r vertices denoted by Pr and a cycle on r vertices is denoted by Cr. K1,m is called a star and it is denoted by Sm. Regular graph on n degree denoted by Qn.
Definition 1.1 A graph labeling is an assignment of integers, to the vertices
or edges, or both of a graph. An edge labeling is a function of E to a set of labels. A graph with such a function defined is called a edge labeled graph.
Definition 1.2 A graph G with p vertices and q edges is a mean graph if there is an injective function f from the edges of G to {0,1,2, ...q}. Such that when each vertex u ∈ V(G) is labeled with
k P n=1
f(en) 2 if
P
f(en) is even and
k P n=1
f(en)+1
2 if
P
f(en) is odd. Where f(en) is the number of edge labels incident with each vertex u ∈ V(G). Then the resulting vertices are all distinct.
Definition 1.3 A graph G with p vertices to be an edge odd mean graph if
there is an injective function f from the edges of G to {1,3,5, ...2q −1}. Such that when each vertex u ∈ V(G) is labeled with
k P n=1
f(en) 2 if
P
f(en) is even and
k P n=1
f(en)+1
2 if
P
f(en) is odd. Where f(en) is the number of edge labels incident with each vertex u∈V(G). Then the resulting vertices are all distinct. Such a function is called an edge odd mean labeling.
Definition 1.4 A graph G with p vertices to be an edge even mean graph if
there is an injective function f from the edges of G to {2,4,6, ...2q}. Such that the vertex labels are given by
k P n=1
f(en)
2 are distinct. Such a function is called an edge even mean labeling.
Definition 1.5 The join of graphs k1 andQn,k1+Qn is obtained by joining
a vertex of k1 with every vertex of Qn with an edge.
Definition 1.6 The corona G1G2 of two graphs G1 and G2 is defined as
the graph G obtained by taking one copy of G (which has p vertices and p copies of G2) and then joining theith vertex of G1 to every vertices in theith copy of G2.
Definition 1.7 The graph [Qn;S2] is obtained from n copies of S2 and the
regular Qn : u1, u2, u3, ...ur by joining uj with the vertex v0 of the jth copy of S2 by means of an edge 1≤j ≤r.
2
Edge Odd And Even Mean Labeling
Theorem 2.1 For n ≥3, the graph Qn has edge odd mean labeling.
proofThe graph Qn has2n vertices and2n−1·n edges. wherenis the number of edges touching with each vertex. Let e1, e2, e3, e4, ...er be the edges of the graph Qn. Define an edge labeling f :E(Qn)→ {1,3,5,7, ...2q−1} by
f(ei) =
(
4i−1 if i= 1,2,3,4, ...r 4i−3 if i= 1,2,3,4, ...r
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph Qn.
Theorem 2.2 For n ≥3, the graph Qn has edge even mean labeling.
proof Define an edge labeling f :E(Qn)→ {2,4,6, ...2q} by f(ei) =
(
4i if i is odd and even 4i−2 if i is odd and even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph Qn.
Theorem 2.3 For n ≥3, the graph K1+Qn has edge odd mean labeling.
proof The graph K1+Qn has 2n+ 1 vertices and 2n−1·n+ 2n edges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph Qn.
Define an edge labeling f :E(K1+Qn)→ {1,3,5, ...2q−1} by f(ei) =
(
4i−3 if i= 1,2,3, ...r 4i−1 if i= 1,2,3, ...r
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph k1+Qn.
Example 2.4 The edge odd mean labeling of K1+Q3 is given in figure1.
t t
t D
D D
D D
D D
figure1
t t t t t t
1 5 9 13 17 21 25
29 373
7
Theorem 2.5 For n ≥3, the graph K1+Qn has edge even mean labeling.
proof Define an edge labeling f :E(K1 +Qn)→ {2,4,6, ...2q} by
f(ei) =
6i−4 if i= 1,2,3, ...r 6i−2 if i= 1,2,3, ...r 3i if i is even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph k1+Qn.
Theorem 2.6 For n ≥3, the graph Pr+Qn has edge odd mean labeling.
proof The graph Pr+Qn has 2n+r vertices and
2n−1·n+r·2n+ (r−1)edges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph Pr+Qn.
Define an edge labeling f :E(Pr+Qn)→ {1,3,5, ...2q−1} by
f(ei) =
(
2i−1 if i is odd 2i−1 if i is even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph Pr+Qn.
Example 2.7 The edge odd mean labeling of Pr+Qn is given in figure2.
t t t t t t t t
t t t
figure 2
1 5 9 13 17 21 25
29 37 33
41 45
3 7
11 15 19 23
27
31 35 39 43
47 51 55
596367
71 75 73 69 65 61
57
53 49
Theorem 2.8 For n ≥3, the graph Pr+Qn has edge even mean labeling.
proof Define an edge labeling f :E(Pr+Qn)→ {2,4,6, ...2q} by
f(ei) =
(
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph Pr+Qn.
Theorem 2.9 For n ≥3, the graph k2+Qn has edge odd mean labeling.
proof The graph K2+Qn has 2n+ 2 vertices and 2n−1·n+ 2n+1+ 1 edges. where n is the number of edges touching with each vertex. Let e be an edge of the graph K2 and let e1, e2, e3, ...er be the edges of the graph Qn.
Define an edge labeling f :E(K2+Qn)→ {1,3,5, ...2q−1} by f(e) = 1
f(ei) =
4i−1 1≤i≤r if r is odd 2i+ 3 i is even
2i+ 3 i is odd if r is odd
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph k2+Qn.
Example 2.10 The edge odd mean labeling of K2+Qn is given in figure3.
s s s s s s s s
s s
figure 3
1 3
5 9 13 17 21 25 29
33 41 45 37
49
11 19 27 35
43 51 53
57 55 47 39 31
23 15 7
Theorem 2.11 For n≥3, the graph K2+Qn has edge even mean labeling.
proof Define an edge labeling f :E(K2 +Qn)→ {2,4,6, ...2q} by f(e) = 2
f(ei) =
4i 1≤i≤r if r is odd 2i+ 4 i is odd
2i+ 4 i is even if r is even
Theorem 2.12 For n ≥3, the graph Qn+Cr has edge odd mean labeling.
proof The graph Qn+Cr has 2n+r vertices and 2n−1·n+r·2n+r edges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph Qn+Cr. Define an edge labeling f :E(Qn+Cr)→
{1,3,5, ...2q−1} by
f(ei) =
(
4i−1 if i= 1,2,3, ...r 4i−3 if i= 1,2,3, ...r
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph Qn+Cr.
Example 2.13 The edge odd mean labeling of Q3+C3 is given in figure4.
t t t t t t t
r r
figure 4
t
t
3 7 11 15 19 23 27
31 35
39 43
47
5 9
13 51 555963 67
71 75 1
21 25
29
33374145 49
53 57 61 65
69 73
77 17
Theorem 2.14 For n ≥3, the graph Qn+Cr has edge even mean labeling.
proof Define an edge labeling f :E(Qn+Cr)→ {2,4,6, ...2q} by
f(ei) =
(
4i−2 if i= 1,2,3, ...r 4i if i= 1,2,3, ...r
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph Qn+Cr.
Theorem 2.15 For n ≥3, the graph QnK1 has edge odd mean labeling.
proof The graph QnK1 has 2n+1 vertices and2n−1·n+ 2n edges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph QnK1.
f(ei) =
6i−5 if i is odd and even 6i−3 if i is odd and even 6i−1 if i is odd and even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph QnK1.
Example 2.16 The edge odd mean labeling of Q3K1 is given in figure5.
s s s s s s s
figure 5
s
s s s
s s s s s
1 3 5 7 9 11 13
15 19 17
21 23
25 27 29 31 33 35 37 39
Theorem 2.17 For n≥3, the graph QnK1 has edge even mean labeling.
proof Define an edge labeling f :E(QnK1)→ {2,4,6, ...2q} by
f(ei) =
6i−4 if i is odd and even 6i−2 if i is odd and even 6i if i is odd and even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph QnK1.
Theorem 2.18 For n ≥3, the graph [Qn;S2] has edge odd mean labeling.
proof The graph [Qn;S2] has 2n+ 2n+1 vertices and 2n−1·n+ 2n+1 edges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph [Qn;S2].
Define an edge labeling f :E([Qn;S2])→ {1,3,5, ...2q−1} by
f(ei) =
(
4i−3 if i is odd and even 4i−1 if i is odd and even
Example 2.19 The edge odd mean labeling of [Q3;S2] is given in figure6.
t t t t t t t t
figure 6
t t t tt t t tt t t t t t tt t
1 5 9 13 17 21 25
29 41 37 33
45
3 7
11 15
19 23
27 31
35 39
43 47
51 55
49 53
Theorem 2.20 For n ≥3,the graph [Qn;S2] has edge even mean labeling.
proof Define an edge labeling f :E([Qn;S2])→ {2,4,6, ...2q} by
f(ei) =
(
4i−2 if i= 1,2,3, ...r 4i if i= 1,2,3, ...r
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph [Qn;S2].
Theorem 2.21 For n, m ≥ 3, the graph Qn K1,m has edge odd mean
labeling.
proofThe graph QnK1,m has2n+m·2nvertices and 2n−1·n+m·2nedges. where n is the number of edges touching with each vertex. Let e1, e2, e3, ...er be the edges of the graph QnK1,m.
Define an edge labeling f :E(QnK1,m)→ {1,3,5, ...2q−1} by
f(ei) =
8i−7 if i= 1,2,3, ...r 8i−5 if i is odd and even 8i−3 if i= 1,2,3, ...r 8i−1 if i is odd and even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge odd mean labeling for the graph Q3K1,3.
Example 2.22 The edge odd mean labeling of Q3K1,3 is given in figure7.
t t t t t t t t
figure 7
t t tt t t t t tt t t t t tt t t t t t t t t
1 9 17 25 33 41 49
57 3 65
11 19
27
35 43
51 59
67 5
13 21
29 37 45
53 61
69 7
15 23
31 39
47 55
71 63
Theorem 2.23 For n, m ≥ 3, the graph Qn K1,m has edge even mean
labeling.
proof Define an edge labeling f :E(QnK1,m)→ {2,4,6, ...2q} by
f(ei) =
8i−6 if i= 1,2,3, ...r 8i−4 if i is odd and even 8i−2 if i= 1,2,3, ...r 8i if i is odd and even
Therefore mean of the vertex labels are all distinct. Hence the function f provides edge even mean labeling for the graph Q3K1,3.
3
Conclusion
The main focus of this paper is resolving the edge mean labeling of a regular graphs for some standard graphs.
4
Acknowledgement
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