Extra Skolem Difference Mean Labeling of Various Graphs

Let graph G=(V(G),E(G)) attains a Skolem difference mean labeling with p vertices and q edges is said to be an extra Skolem difference mean labeling of graph G if all the labels of the vertices are odd. The graph which attains an extra Skolem difference mean labeling is called an extra Skolem difference mean graph. We obtain an extra Skolem difference mean labeling for Comb graph, Twig of a path P_n, H graph of a path P_n, K_1,2*K_(1,n) graph, K_1,3*K_(1,n) graph, m- Join of H_n, P_n ⊙ K_(1,m) graph , HSS(P_n) graph, H ⊙ 〖 mK 〗 _1-graph of a path P_n.


Introduction
We consider finite, connected and undirected graph. We consider graph having set of vertices ( ) and set of edges ( ). An excellence reference on this subject is the survey by J. A. Gallian [4]. Skolem difference mean labeling was introduced by K. Murugan and A. Subramanian in [6]. Selvi, Ramya and Jeyanthi [7] define an extra Skolem difference mean labeling of graphs. We refer Gross and Yellen [5], for all kinds of definitions and notations. , if | ( ) − ( )| is odd and the graph is called a Skolem difference mean graph . [1] Definition: -Let graph = ( ( ), ( )) attains a Skolem difference mean labeling with vertices and edges is said to be an extra Skolem difference mean labeling of graph if all the labels of the vertices are odd. The graph which attains an extra Skolem difference mean labeling is called an extra Skolem difference mean graph. [2] SOME EXISTING RESULTS:
Definition: , * , is the graph obtained from 1,3 by attaching root of a star 1, at each pendant vertex of 1,3 . [1] Definition: The corona ⊙ of two graphs 1 & 2 is defined as the graph obtained by taking one copy of 1 (which has 1 vertices) and 1 copies of 2 and joining the ℎ vertex of 1 to every vertices in the ℎ copy of 2 . [10] Definition: −Joins of graph is a graph where each of graph denoted by 1 by an edge 1 with graph denoted by 2 , graph denoted by 2 by an edge 2 with graph denoted by 3 and so on with graph denoted by −1 by an edge −1 with graph denoted by such that 1 = 2 = . [9] Definition: Let be a graph. A graph obtained from by replacing each edge by a H graph in such a way that the ends of are merged with a pendant vertex in 2 and pendant vertex in ′ 2 is called super subdivision of is denoted by ( ), where the graph is a tree on 6 vertices in which exactly two vertices of degree 3.
Which is bijective function. Hence Comb ⊙ 1 graph is an extra Skolem difference mean graph.
Illustration: An extra Skolem difference mean labeling of 5 ⊙ 1 is shown in Figure-1.
Which is bijective function. So, ( ) is an extra Skolem difference Mean graph.
Case:2 If is even

is even
Which is bijective function.
Hence −graph of a path graph is an extra Skolem difference mean graph.
Case:1 is even.
Hence the graph ( ) is an extra Skolem difference mean graph.
Illustration: An extra Skolem difference mean labeling of ( 4 ) is shown in Figure-8. Hence the graph ⊙ 1 graph of a path is an extra Skolem difference mean graph.
Illustration: An extra Skolem difference mean labeling of ⊙ 2 1 graph of a path 5 is shown in Figure-9.

Conclusion
In this paper we obtain an extra Skolem difference mean labeling for Comb graph, Twig of a path , graph of a path , 1,2 * 1, graph, K 1,3 * K 1,n graph, − Join of , ⊙ 1, graph , ( ) graph, ⊙ 1 −graph of a path .We can discuss more similar results for various graphs.