On (distance) Laplacian characteristic polynomials of power graphs

TL;DR

This paper derives the characteristic polynomials of Laplacian and distance Laplacian matrices for power graphs of certain finite groups, providing explicit formulas and inequalities for their zeros.

Contribution

It presents explicit formulas for characteristic polynomials of Laplacian matrices of power graphs of groups of order pqr and cyclic/dicyclic groups, along with inequalities for their zeros.

Findings

Characteristic polynomials for power graphs of groups of order pqr are obtained.

Characteristic polynomials for proper power graphs of cyclic and dicyclic groups are provided.

Inequalities for zeros of the distance Laplacian characteristic polynomials are discussed.

Abstract

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order $ pqr $, where $ p,q $ and $ r $ are { primes,} are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.