Integral Sum Graphs from Complete Graphs Cycles and Wheels PDF

Keywords

integers, vertices, sum graph

Abstract

A graph G is an integral sum graph if there is a labelling θ 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 θ(u) + θ(v) = θ(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 σ(G) such that G ∪ σ (G) K| is a sum graph, and there is a minimum number ζ (G) such that G ∪ ζ (G)K| is an integral sum graph. In this paper, we prove a conjecture of Harary that ζ(Kn) = σ (Kn) for all Kn with n ≥ 4. Also, we show that cycles Cn and wheels Wn are integral sum graphs or all n ≠ 4