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

TL;DR

This paper introduces the $m$-graph $m-G(H)$ for a finite abelian group $H$, where two vertices $a$ and $b$ are adjacent if $b=a^m$ or $a=b^m$, and analyzes its connectivity, structure, and diameter. It establishes a precise connectivity criterion in terms of prime divisors via $k=\gcd(m,n)$, showing that connected graphs are trees and are graph-isomorphic to $k$-G$(H)$. For cyclic $H=Z_n$, it gives explicit degree formulas, diameters in terms of the least $w$ with $k^w\equiv 0 \pmod{n}$, and characterizes several special cases including when $n\mid m$ (yielding $K_{1,n-1}$). For non-cyclic abelian groups, it extends the analysis through a product-graph framework $(d_1,\dots,d_i)$-PG$(H)$, proving connected $m$-G$(H)$ are trees and deriving diameter bounds; the results unify the tree-structure across cyclic and non-cyclic types and include constructive examples. The work provides a cohesive treatment of how abelian group structure governs the connectivity, tree-ness, and metric properties of the associated $m$-graphs, with potential implications for spectral and combinatorial properties tied to these graphs.

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.

Theorems & Definitions (54)

• Remark 1.1
• Example 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Corollary 2.4
• proof
• Example 2.5
• Lemma 2.6
• proof