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

TL;DR

This paper introduces the concept of the $m$-graph of a finite abelian group, exploring its properties and structural characteristics based on the group's algebraic structure.

Contribution

It defines the $m$-graph for finite abelian groups and investigates its properties, providing new insights into the interplay between group theory and graph theory.

Findings

Characterization of the $m$-graph structure

Conditions for connectivity of $m$-graphs

Relationships between group properties and graph features

Abstract

Let $H$ be a finite abelian (commutative) group of order $n \geq 2$, 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.