Universal adjacency spectrum of (proper) power graphs and their complements on some groups

TL;DR

This paper investigates the universal adjacency spectra of power graphs and their complements on specific groups, providing new spectral results and unifying various graph matrices in a group-theoretic context.

Contribution

It introduces the spectral analysis of universal adjacency matrices for power graphs and their complements on certain groups, extending known results and unifying different graph matrices.

Findings

Spectral results for power graphs on groups like
7e_n, D_n, Q_n.

Full spectrum determination in specific cases.

New insights into the spectra of graph complements on groups.

Abstract

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple graph $H$ with adjacency matrix $A(H)$ and degree diagonal matrix $D(H)$, the universal adjacency matrix is $U(H)= \alpha A(H)+\beta D(H)+ \gamma I +\eta J$, where $\alpha (\neq 0), \beta, \gamma, \eta \in \mathbb{R}$, $I$ is the identity matrix and $J$ is the all-ones matrix of suitable order. One can study many graph-associated matrices, such as adjacency, Laplacian, signless Laplacian, Seidel etc. in a unified manner through the universal adjacency matrix of a graph. Here we study universal adjacency eigenvalues and eigenvectors of power graphs, proper power graphs and their complements on the group $\mathbb{Z}_n$, dihedral group ${D}_n$, and the generalized quaternion group ${Q}_n$. Spectral results of no kind for the complement of power graph on any group were obtained before. We determine the full spectrum in some particular cases. Moreover, several existing results can be obtained as very specific cases of some results of the paper.