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 \(m \geq |S|+1\) (\(m \geq |S|\)). 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.