Some results on exclusive sum graphs

Abstract

Let N(Z) denote the set of all positive integers (integers). The sum graph G+(S) of a finite subset S⊂N(Z) is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S⊂N(Z). A sum labelling S is called an exclusive sum labelling if u+v∈S∖V(G) for any edge uv∈E(G). We say that G is labeled exclusively. The least number r of isolated vertices such that G∪rK1 is an exclusive sum graph is called the exclusive sum number e(G) of graph G. In this paper, we discuss the exclusive sum number of disjoint union of two graphs and the exclusive sum number of some graph classes.