On the Spectral Analysis of Power Graph of Dihedral Groups

TL;DR

This paper analyzes the spectral properties of power graphs of dihedral groups, providing explicit spectra calculations and correcting previous results, thereby expanding understanding of these graphs' algebraic and spectral characteristics.

Contribution

It determines the adjacency, Laplacian, and signless Laplacian spectra of power graphs of dihedral groups and corrects prior assumptions about their spectral properties.

Findings

Spectra are explicitly calculated for dihedral groups with order 2pq.

Previous results are shown to be valid only for prime power orders.

The study provides counterexamples to earlier claims and refines the spectral analysis of power graphs.

Abstract

The power graph \( \mathcal{G}_G \) of a group \( G \) is a graph whose vertex set is \( G \), and two elements \( x, y \in G \) are adjacent if one is an integral power of the other. In this paper, we determine the adjacency, Laplacian, and signless Laplacian spectra of the power graph of the dihedral group \( D_{2pq} \), where \( p \) and \( q \) are distinct primes. Our findings demonstrate that the results of Romdhini et al. [2024], published in the \textit{European Journal of Pure and Applied Mathematics}, do not hold universally for all \( n \geq 3 \). Our analysis demonstrates that their results hold true exclusively when \( n = p^m \) where \( p \) is a prime number and \( m \) is a positive integer. The research examines their methodology via explicit counterexamples to expose its boundaries and establish corrected results. This study improves past research by expanding the spectrum evaluation of power graphs linked to dihedral groups.