Integral sum numbers of graphs

Abstract

The sum graph of a set S of positive integers is the graph G(+)(S) having S as its vertex set, with two distinct vertices adjacent whenever their sum is in S. If S is allowed to be a subset of all integers, a graph so obtained is called an integral sum graph. The integral sum number of a given graph G is the smallest number of isolated vertices which when added to G result in an integral sum graph. In this paper, we find the integral sum numbers of caterpillars, cycles, wheels, and complete bipartite graphs.