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

## Authors

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.

## Abstract

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.