arXiv subjects

Basit Auyoob Mir

Publications and source records attributed to Basit Auyoob Mir.

3 recordsLinked to original sources

arXiv2026

Spectral radius and second largest eigenvalues of power graphs of finite groups

Consider a group $\mathbb{G}$ and construct its power graph, whose vertex set consists of the elements of $\mathbb{G}$. 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 this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group $C_{n}$, the dihedral group $\mathcal{D}_{2n}$, and the dicyclic group $\mathcal{Q}_{4n}$. For $n\neq p^{m},$ the power graph of the cyclic group $C_{n}$ is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups

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arXiv2025

On the Spectral Analysis of the Superpower Graph of the Direct Product of Dihedral Groups

The superpower graph of a finite group $G$, or $\mathcal{S}_G$, is an undirected simple graph whose vertices are the elements of the group $G$, and two distinct vertices $a,b\in G$ are adjacent if and only if the order of one vertex divides the order of the other vertex, which means that either $o(a)|o(b)$ or $o(b)|o(a)$. In this paper, we have investigated the $A_α$-adjacency spectral properties of the superpower graph of the direct product $D_p\times D_p$, where $D_p$ is a dihedral group for $p$ being prime. Also, we have determined its Laplacian and signless Laplacian spectrum by giving different values to $α$; furthermore, we delved into its superpower graph and deduced the $A_α$- adjacency spectrum of the superpower graph of $D_p\times D_p$ and $D_{p^m}$ for $p$ being an odd prime.

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arXiv2025

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 \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.

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