On the Spectral Analysis of Power Graph of Dihedral Groups

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 $\mathit{\text{European}}\mathit{\text{Journal}}\mathit{\text{of}}\mathit{\text{Pure}}\mathit{\text{and}}\mathit{\text{Applied}}\mathit{\text{Mathematics}}$, do not hold universally for all $n\ge 3$. 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.