The Mod Sum Number of Even Fans and Symmetric Complete Bipartite Graphs

Abstract

A graph \(G = (V, E)\) is a mod sum graph if there exists a positive integer \(z\) and a labeling, \(\lambda\), of the vertices of \(G\) with distinct elements from \(\{1, 2, \ldots, z-1\}\) such that \(uv \in E\) if and only if the sum, modulo \(z\), of the labels assigned to \(u\) and \(v\) is the label of a vertex of \(G\). The mod sum number \(\rho(G)\) of a connected graph \(G\) is the smallest nonnegative integer \(m\) such that \(G \cup mK_1\), the union of \(G\) and \(m\) isolated vertices, is a mod sum graph. In Section \(2\), we prove that \(F_n\) is not a mod sum graph and give the mod sum number of \(F_n\) (\(n \geq 6\) is even). In Section \(3\), we give the mod sum number of the symmetric complete graph.