On integral sum graphs

A graph is said to be a sum graph if there exists a set S of positive integers as its node set, with two nodes adjacent whenever their sum is in S. An integral sum graph is defined just as the sum graph, the difference being that S is a subset of Z instead of N∗. The sum number of a given graph G is defined as the smallest number of isolated nodes which when added to G result in a sum graph. The integral sum number of G is analogous. We are interested in some problems and conjectures posed by Harary (1994) in [4], mainly proving the conjecture that the integral sum numbers equal sum numbers for all complete graphs with at least four nodes and disproving the conjecture that every tree T with integral sum number ζ (T) = 0 is a caterpillar. In addition, we also generalize some results of [4].