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 $\{1,2,\dots ,z-1\}$ 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 $G\cup m{K}_1$, 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 $F_n$ ($n\ge 6$ is even). In Section $3$, we give the mod sum number of the symmetric complete graph.