# Laplacian spectra of power graphs of certain finite groups

## TL;DR

This paper investigates the Laplacian spectra of power graphs of specific finite groups, providing complete characterizations of algebraic connectivity, spectral radius multiplicities, and Laplacian integrality for these groups.

## Contribution

It offers new comprehensive results on the Laplacian spectra of power graphs of cyclic, dicyclic, and finite p-groups, including conditions for Laplacian integrality and spectral properties.

## Key findings

- Algebraic connectivity determined for finite p-groups
- Laplacian spectral radius multiplicity characterized for dicyclic and p-groups
- Power graphs of finite p-groups are Laplacian integral

## Abstract

In this article, various aspects of Laplacian spectra of power graphs of finite cyclic, dicyclic and finite $p$-groups are studied. Algebraic connectivity of power graphs of the above groups are considered and determined completely for that of finite $p$-groups. Further, the multiplicity of Laplacian spectral radius of power graphs of the above groups are studied and determined completely for that of dicyclic and finite $p$-groups. The equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above groups. Orders of dicyclic groups, for which their power graphs are Laplacian integral, are determined. Moreover, it is proved that the notion of equality of the vertex connectivity and the algebraic connectivity and the notion of Laplacian integral are equivalent for power graphs of dicyclic groups. All possible values of Laplacian eigenvalues are obtained for power graphs of finite $p$-groups. This shows that power graphs of finite $p$-groups are Laplacian integral.