The Sum Numbers and The Integral Sum Numbers of \(\overline{C_n}\) and \(\overline{W_n}\)

Abstract

Let \(G = (V,E)\) be a simple graph. \({N}\) and \({Z}\) denote the set of all positive integers and the set of all integers, respectively. The sum graph \(G^+(S)\) of a finite subset \(S \subset{N}\) is the graph \((S, {E})\) with \(uv \in {E}\) if and only if \(u+v \in S\). \(G\) is a sum graph if it is isomorphic to the sum graph of some \(S \subseteq {N}\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices, which result in a sum graph when added to \(G\). By extending \({N}\) to \({Z}\), the notions of the integral sum graph and the integral sum number of \(G\) are obtained, respectively. In this paper, we prove that \(\zeta(\overline{C_n}) = \sigma(\overline{C_n}) = 2n-7\) and that \(\zeta(\overline{W_n}) = \sigma(\overline{W_n}) = 2n-8\) for \(n \geq 7\).