Structural and Spectral Properties of Prime Order Element Graph of Finite Abelian Groups

TL;DR

This paper analyzes the structure and spectral properties of Prime Order Element (POE) graphs of finite Abelian groups, revealing how group order influences graph connectivity, components, and eigenvalues.

Contribution

It provides a comprehensive structural and spectral analysis of POE graphs for finite Abelian groups, including explicit eigenvalue calculations and graph component characterizations.

Findings

POE graph is connected if group order is square-free

POE graphs of non-square-free groups have multiple components

Eigenvalues are explicitly derived for various graph structures

Abstract

Given a finite group $G$, the \emph{Prime Order Element (POE) Graph} $\Gamma(G)$ consists of the group elements as the vertices, and two vertices $x$ and $y$ are adjacent if and only if $o(xy)$ is prime. This paper presents a thorough structural and spectral analysis of the POE graphs associated with the finite Abelian groups of different types. The order of a finite Abelian group may be a prime or a product of primes, which influences the structure of POE graphs. The POE graph is connected when the order of the Abelian group is a square-free integer. The POE graphs of the other Abelian groups have multiple connected components. Some of these components are isomorphic to the POE graph of a lower-order group. We study various graph-theoretic properties of the components, including regularity and bipartiteness. Arranging the elements of the group in a number of particular orders, we observe the block structure in the adjacency matrix of POE graphs. It assists us in investigating the spectral properties of POE graphs. We explicitly derive the characteristic polynomials governing both integral and irrational eigenvalues, and compute the eigenvalues with multiplicity in terms of the structure of the graphs.