Bounds on the number of isolates in sum graph labeling

Abstract

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L( w)= L( u)+ L( v). The sum number σ( G) of a graph G=( V, E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that σ( G)⩽| E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G=( V, E) with fixed | V|⩾3 and | E|, the average of σ( G) is at least |E|−3|V|( log|V|)/[ log(( |V| 2 )/|E|)]−|V|−1 . In other words, for most graphs, σ(G)∈Ω(|E|).