On the $m$-graph of a finite Abelian Group

Abstract

Let $H$ be a finite abelian (commutative) group of order $n\ge 2$, and $m>1$ be an integer. We define the $m$-graph of $H$, denoted by $m-G(H)$, as a simple undirected graph with vertex set $H$, and two distinct vertices, $a,b\in H$, are connected by an edge if and only if $a^m=b$ or $b^m=a$. Several results regarding the properties of the $m$-$G(H)$ have been established.