Search

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., Comment: 9 pages, 1 figures more...

We consider the group G and construct its power graph, whose vertex set consists of the elements of G. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In the paper (Indagationes Mathematicae 29(2) (2018), pp 730 t0 737), Chattopadhyay et al. gave spectral radius bounds of the power graph of certain finite groups. In this article, we improved the bounds of the spectral radius of the power graphs of the above results. Furthermore, we provide bounds for the distance spectral radius of the power graph of the cyclic group Cn, the dihedral group D2n, and the dicyclic group Q4n. For some cases, we find the bounds are exact if and only if they pertain to a particular family of graphs. more...