The Sum Numbers and the Integral Sum Numbers of the Graph $K_n\text{E}(C_{n-1})$

Abstract

The concept of the sum graph and integral sum graph were introduced by F. Harary. Let $\mathbb{N}$ denote the set of all positive integers. 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 $u+v\in S$. A simple graph $G$ is said to be a sum graph if it is isomorphic to a sum graph of some $S\subset N$. The sum number $\sigma (G)$ of $G$ is the smallest number of isolated vertices which when added to $G$ result in a sum graph. Let $\mathbb{Z}$ denote the set of all integers. The integral sum graph $G^{+}(S)$ of a finite subset $S\subset Z$ is the graph $(S,E)$ with $uv\in E$ if and only if $u+v\in S$. A simple graph $G$ is said to be an integral sum graph if it is isomorphic to an integral sum graph of some $S\subset Z$. The integral sum number $\zeta (G)$ of $G$ is the smallest number of isolated vertices which when added to $G$ result in an integral sum graph. In this paper, we investigate and determine the sum number and the integral sum number of the graph $K_n\setminus E(C_{n-1})$. The results are presented as follows:$\zeta (K_n\setminus (C_{n-1}))=\{\begin{array}{ll}0,& n=4,5,6,7\\ 2n-7,& n\ge 8\end{array}$
and
$\sigma (K_n\setminus E(C_{n-1}))=\{\begin{array}{ll}1,& n=4\\ 2,& n=5\\ 5,& n=5\\ 7,& n=7\\ 2n-7,& n\ge 8\end{array}$