Some Results on mod (integral) Sum Graphs

Abstract

Let $N(Z)$ denote the set of all positive integers (integers). The sum graph $G_S$ of a finite subset $S\subset N(Z)$ is the graph $(S,E)$ with $uv\in E$ if and only if $u+v\in S$. A graph $G$ is said to be an (integral) sum graph if it is isomorphic to the sum graph of some $S\subset N(Z)$. The (integral) sum number $\sigma (G)$ of $G$ is the smallest number of isolated vertices which when added to $G$ result in an (integral) sum graph. A mod (integral) sum graph is a sum graph with $S\subset Z_m\setminus \{0\}$ ($S\subset Z_m$) and all arithmetic performed modulo $m$ where $m\ge |S|+1$ ($m\ge |S|$). The mod (integral) sum number $\rho (G)$ of $G$ is the least number $\rho$ ($\psi$) of isolated vertices $\rho K_1$ ($\psi K_1$) such that $G\cup \rho K_1$ ($G\cup \psi K_1$) is a mod (integral) sum graph. In this paper, the mod (integral) sum numbers of $K_{r,s}$ and $K_n–E(K_r)$ are investigated and bounded, and $n$-spoked wheel $W_n$ is shown to be a mod integral sum graph.