Distance matrix of enhanced power graphs of finite groups

TL;DR

This paper computes the spectra of the distance matrix of enhanced power graphs for various finite groups, including non-abelian, dihedral, dicyclic, and certain abelian groups, revealing their spectral properties.

Contribution

It provides explicit spectral computations for the distance matrix of enhanced power graphs of several classes of finite groups, extending understanding of their algebraic graph properties.

Findings

Spectra of the distance matrix for non-abelian groups of order pq

Spectra for dihedral and dicyclic groups

Spectral results for elementary abelian and non-cyclic abelian groups

Abstract

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.