On Construction of Mean Graphs

By a graph we mean a finite, simple and undirected one. The vertex set and the edge set of a graph G are denoted by V(G) and E(G) respectively. The disjoint union of m copies of the graph G is denoted by mG. The union of two graphs G1 and G2 is the graph G1 ∪ G2 with V(G1 ∪ G2) = V (G1) ∪ V (G2) and E (G1 ∪ G2) = E(G1) ∪ E(G2) . A vertex of degree one is called a pendant vertex. Let ) , ( q p G = be a mean graph with p vertices and q edges and let v be a vertex with label q and let one of the mean labelings of G satisfy the following: If q is odd (even) and all the labels of the vertices w hich are adjacent to v are even (odd), then we call this mean labeling as extra mean labeling [4] and the graph G as extra mean graph.

q such that for each edge uv, is labeled Then the resulting edge labels are distinct.In this paper, we prove some general theorems on mean graphs and show that the graphs , Jewel graph

Introduction
By a graph we mean a finite, simple and undirected one.The vertex set and the edge set of a graph G are denoted by V(G) and E(G) respectively.The disjoint union of m copies of the graph G is denoted by mG.The union of two graphs G 1 and G 2 is the graph . A vertex of degree one is called a pendant vertex.Let ) , ( q p G = be a mean graph with p vertices and q edges and let v be a vertex with label q and let one of the mean labelings of G satisfy the following: If q is odd (even) and all the labels of the vertices which are adjacent to v are even (odd), then we call this mean labeling as extra mean labeling [4] and the graph G as extra mean graph.
. Terms and notations not defined here are used in the sense of Harary [1].
The concept of mean labeling was introduced by Somasundaram and Ponraj [2] and further studied by the same authors in [3].Motivated by the work of the above authors, we have established the mean labeling of some standard graphs in [4,5].In this paper we extend our study to establish the mean labeling some more graphs like Jewel graph n J and Jelly fish graph n JF) ( .

Mean Graphs
Remark 2.1: For any mean graph G, q q and 1 , 0 − must be the vertex labels.Either 1 or 2 must be a vertex labeling, a vertex of label q -1 is adjacent with a vertex of label q and a vertex of label 0 is adjacent with a vertex of label 1 or 2.
. Now the label of the edge xy is . Also the label of the edge uv is 2 2 and the label of the edge uv is 2 2 Hence the edge labels of the graph G are 1, 2, 3,…, 2

+ + q q
and the vertex labels of G are also distinct.This completes the proof.

Example 2.3:
.The mean labeling of G 1 and G 2 are given below.The mean graph obtained by the above construction is given in Fig. 1.

Fig. 1
Theorem 2.4: Let ) , ( be a mean graph with mean labeling f and let ux e = be an edge with 1 ) e with the edge e (that is identifying u with v and x with y), then G is a mean graph.
Hence G is a mean graph.The mean graph obtained by the above construction is given Fig. 2. G obtained by identifying the edge ' e with the edge e (that is identifying x with y and u with v), then G is a mean graph.

Example 2.5: Let
. Clearly G has G is a mean graph, the vertex labels of 1 G under h are remain distinct and G is a mean graph, the labels of the vertices of } , { ) . The edge labels of the graph 1 G , except the edges incident with u, under h remain distinct.Since 1 G is an extra mean graph with mean labeling f, for each vertex w incident with u in G 1 , ) (u f and ) (w f are of opposite parity.Therefore the induced edge label under f is , 2 Hence, the induced edge labels of 1 q and the edge labels of 2

− +q q
Hence G is a mean graph.The mean graph obtained by the above construction is given in Fig. 3. Fig. 3 Theorem 2.8: The Jewel graph n J is an extra mean graph.
For each vertex label f , the induced edge label * f is defined as follows: Clearly f is a mean labeling of G. Moreover q is odd and all the vertices which are adjacent to the vertex labeled q are even.Thus, G is an extra mean graph.
Example 2.9: The mean labeling of 5 J is given in Fig. 4. Theorem 2.10: and the edge set }. Then G is a mean graph.
For each vertex label f , the induced edge label * f is defined as follows: It can be verified that f is a mean labeling of G. Hence G is a mean graph.
Example 2.11: The mean labeling of 5 9 ) ( K P + is given in Fig. 5.

Fig. 5
Theorem 2.12: The graph Jelly fish n JF) ( is a mean graph.
For each vertex label f , the induced edge label * f is defined as follows:

.
If G is a graph obtained by identifying the edge ' .The mean labeling of 1 G and 2 G are given below. Fig.2

G and a mean labeling of 2 G
The extra mean labeling of 1 are given below. Fig.4

Example 2 .
15: Let G be a Comb obtained from the path 4 P .The mean labeling of ) (+ G is given in Fig. 7.

Theorem 2. 2: Let
the vertex x with y and u with v by an edge, then G is a mean graph.
is given in Fig.6.
G is a mean graph.