∆-Optimum Exclusive sum labeling of Book graph (Bn)


Sheetal Thakare, Dept. of App. Mathematics, Lokmanya Tilak College of Engg., Navi Mumbai,India.
Manisha Acharya, Associate Professor, Dept. of Maths, M.D.College of Arts,Commerce and Science, Parel, Mumbai,India.


A graph G = (V, E) is called sum graph if there exists an injective function L called sum labeling, from V to a set of positive integers such that for x, y ∈ V, xy ∈ E if and only if L(x) + L(y) = L(w) for some vertex w in V . In such case w is called working vertex of graph G. A sum labeling of a graph G∪Kr for some positive integer r is said to be exclusive with respect to G if all its working vertices are in Kr. In order to get lebeled exclusively; every graph G needs some isolated vertices. The smallest number of such isolates that need to be added to a graph G is called the exclusive sum number of the graph G; denoted by (G). The number of isolates must be at least ∆(G), the maximum vertex degree in G. In case (G) = ∆(G), then G is said to be a ∆-optimum exclusive sum graph and the exclusive sum labeling of G using ∆(G) isolates is called ∆-optimum exclusive sum labeling of G. In this paper we find exclusive sum number of Book graph, Bn and show that it is a ∆-optimum exclusive sum graph. We also define a ∆-optimum exclusive sum labeling for Book graph.