On the Spectral Analysis of Power Graph of Dihedral Groups
Basit Auyoob Mir, Fouzul Atik, Priti Prasanna Mondal
TL;DR
The paper addresses the spectral analysis of the power graph of the dihedral group D_{2pq} with distinct primes p and q, testing the universality of Romdhini et al.'s previous formulas. Using a block-structure decomposition, it derives the adjacency spectrum via a quintic for the remaining eigenvalues and provides closed-form Laplacian and signless Laplacian spectra with explicit coefficient expressions. The main contributions are corrected spectral formulas for D_{2pq}, including explicit eigenvalues 0^{(pq-1)} and -1^{(pq-4)} for the adjacency spectrum, along with detailed Laplacian and signless Laplacian spectra, and a concrete counterexample showing Romdhini's results fail unless n is a prime power. This work advances power-graph spectral theory for non-abelian groups and offers a methodological framework that extends to other group classes and potential applications in network dynamics.
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.