On the Spectral properties of power graphs over certain groups

TL;DR

This paper analyzes the spectral properties of the power graph of a specific group, providing explicit calculations of its characteristic polynomial, spectrum, and energy, thus advancing understanding of algebraic graph spectra.

Contribution

It explicitly determines the power graph of a particular group and computes its spectral properties, including characteristic polynomial, spectrum, and Laplacian energy, which is a novel contribution.

Findings

Explicit power graph of the given group is determined.

Characteristic polynomial for adjacency, Laplacian, and signless Laplacian matrices is computed.

Spectral properties and Laplacian energy of the power graph are derived.

Abstract

The power graph $P(\Omega)$ of a group $\Omega$ is a graph with the vertex set $\Omega$ such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph of the group $\mathcal{G} = \langle s,r \, : r^{2^kp} = s^2 = e,~ srs^{-1} = r^{2^{k-1}p-1}\rangle$. Further, we compute its characteristic polynomial for the adjacency, Laplacian, and signless Laplacian matrices associated with this power graph. In addition, we determine its spectrum, Laplacian spectrum, and Laplacian energy.