A Theorem on Sum Graphs

A graph G is a sum graph if there exists a labeling of the vertices of G by distinct positive integers such that the vertices labeled u and v are adjacent if and only if there exists a vertex labeled u + v. The sum number σ(G) of a graph G is defined as the smallest number of isolated vertices which, when added to G results in a sum graph. A labeling that realizes G∪ σ(G)K1 as a sum graph is said to be optimal. In this paper we provide for the first time an optimal sum graph labeling of the crowns Cn ⊙ K1(n ≥ 3) whose smallest label is 1 and also show that σ (Cn ⊙ K1) = 1.