Abstract
A graph G is an integral sum graph if there is a labelling theta of its vertices with distinct integers, so that for any two distinct vertices u and v, uv is an edge of G if and only if theta(u) + theta(v) = theta(w) for some vertex w. G is a sum graph if the labels are positive integers. For each graph G there is a minimum number sigma(G) such that G boolean OR sigma (G)K-1 is a sum graph, and there is a minimum number zeta(G) such that G boolean OR zeta(G)K-1 is an integral sum graph. In this paper, we prove a conjecture of Harary that zeta(K-n) = sigma(K-n) for all K-n with n greater than or equal to 4. Also, we show that cycles C-n and wheels W-n are integral sum graphs for all n not equal 4.