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 $n\ge 7$.