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

TL;DR

This paper investigates the spectral properties of power graphs derived from finite groups, providing bounds and exact values for spectral radii and eigenvalues, with implications for graph structure and group theory.

Contribution

It improves bounds on spectral radii and eigenvalues of power graphs for specific finite groups and characterizes when these bounds are tight.

Findings

Bounds for spectral radius of power graphs of cyclic, dihedral, and dicyclic groups.

Second largest eigenvalue bounds related to clique number.

Exact bounds identified for certain graph families.

Abstract

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