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

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:\(\zeta(K_n \setminus (C_{n-1})) = \begin{cases}
0, & n = 4,5,6,7 \\
2n-7, & n \geq 8
\end{cases}\)
and
\(\sigma(K_n \setminus E(C_{n-1})) = \begin{cases}
1, & n = 4 \\
2, & n = 5\\
5, & n = 5\\
7, & n = 7\\
2n-7, & n \geq 8
\end{cases}\)