Vertex Equitable Labeling of Cycle and Star Related Graphs

→ A induces an edge labeling f* defined by f*(uv) = f(u) + f(v) for all edges uv. For a A ∈ , let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, ( ) ( ) 1 v a v b f f − ≤ and the induced edge labels are 1, 2, 3,..., q. In this paper, we prove that jewel graph Jn , jelly fish graph (JF)n, balanced lobster graph BL(n,2,m), Ln m K and 1, ˆ n m L oK are vertex equitable graphs.


Introduction
All graphs considered here are simple, finite, connected and undirected.We follow the basic notations and terminologies of graph theory as in [1].A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions.There are several types of labeling and a detailed survey of graph labeling can be found in [2].The vertex set and the edge set of a graph are denoted by V(G) and E(G) respectively.The concept of vertex equitable labeling was due to Lourdusamy and Seenivasan [3] and further studied in [4][5][6][7].This is the further extension work on vertex equitable labeling and we prove that the jewel graph J n , jelly fish graph (JF) n , the lobster L(n,2,m), L n m K and The jelly fish graph (JF) n is a graph with vertex set

Definition 1.3:
The corona G 1 G 2 of the graphs G 1 and G 2 is defined as a graph obtained by taking one copy of G 1 (with p vertices) and p copies of G 2 and then joining the i th vertex of G 1 to every vertex of the i th copy of G 2 .

Definition 1.4:
A tree, which yields a path when its pendant vertices are removed, is called a caterpillar.A tree, which yields a caterpillar when its pendant vertices are removed, is called a lobster.Let L(n, 2, k) be the lobster constructed as follows.Let a 1 , a 2 ,…,a n be the vertices of the path P n and a i1 and a i2 be the vertices adjacent to a i , 1≤i≤ n.Join a ij with the pendant vertices a ij is the graph obtained from ladder L n and 2n copies of K 1,m by identifying a non central vertex of i th copy of K 1,m with i th vertex of L n .

Main Results
Theorem 2.1: Let G 1 (p 1 , q 1 ), G 2 (p 2 , q 2 ) ,…, G m (p m , q m ) be a vertex equitable graphs with vertex equitable labeling f i where q i (1 ) i m ≤ ≤ is even and let ' be an edge with ( ) and ( ) 0, ( ) 1 . Define a vertex labeling f: V (G) → A as follows. ( The edge labels of the graph G 1 will remain fixed, the edge labels of the graph G i (2 . The bridges between the two graphs G i ,

Example 1:
The vertex equitable labeling of G 1 , G 2 and G 3 are given below: The vertex equitable labeling of the graph obtained by the above construction is given in Fig. 1.Theorem 2.2: Let G 1 (p 1 , q 1 ), G 2 (p 2 , q 2 ) ,…, G m (p m , q m ) be a vertex equitable graphs with vertex equitable labeling f i where q i (1 ) i m ≤ ≤ is odd and let ' be an edge with ( ) , ( ) The edge labels of the graph G 1 will remain fixed, the edge labels of the graph G i (except

Example 2:
The vertex equitable labeling of G 1 , G 2 and G 3 are given below: The vertex equitable labeling of the graph obtained by the above construction is given in Fig. 2.
Fig. 2 Theorem 2.3: The jewel graph J n is a vertex equitable graph.

Proof:
Let vertex set
. Define a vertex labeling f: V (G) → A as follows. ) It can be verified that the induced edge labels of J n are 1, 2,…, 2n+5 and j∈A.Clearly f is a vertex equitable labeling of J n .Thus, Jewel graph J n is a vertex equitable graph.
( ) 0, ( ) 1 2, . It can be verified that the induced edge labels of (JF) n are 1, 2,…, 2n+1 and ≤ for all i, j∈A.Clearly f is a vertex equitable labeling of (JF) n .Thus, Jelly fish graph (JF) n is a vertex equitable graph.

Example 4:
The vertex equitable labeling of (JF) 6 is given in Fig. 4.
Thus, L n K m is a vertex equitable graph.

Definition 1 . 1 :
are vertex equitable.The following definitions have been used in the subsequent section.The jewel graph J n is a graph with vertex setV(J n )={ , , ,and edge set E(J n )= { , , , , , , ux uy xy xv yv uu vu i

≤Example 7 :
Fig.7 a graph obtained by identifying x i with