The Laplacian spectrum of power graphs of some finite abelian p-groups

TL;DR

This paper studies the Laplacian spectrum of power graphs of finite abelian p-groups, revealing spectral properties and relationships between group spectra and graph spectra, especially for planar cases.

Contribution

It establishes that the group spectrum is contained within the Laplacian spectrum of its power graph for certain finite abelian p-groups and planar groups.

Findings

Spectrum of $bZ_{p^m}^n$ is contained in the Laplacian spectrum of its power graph.

For planar finite abelian groups, the group spectrum is contained in the Laplacian spectrum.

Provides spectral characterizations for power graphs of specific finite abelian groups.

Abstract

The power graph $\mathcal{G}(G)$ of a group $G$ is a simple graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if one is a power of other. In this paper, we investigate the Laplacian spectrum of the power graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$ of finite abelian $p$-group $\mathbb{Z}_{p^m}^n$. In particular, we prove that the spectrum of group $\mathbb{Z}_{p^m}^n$ is contained in the Laplacian spectrum of graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$. For a finite abelian group $G$ whose power graph $\mathcal{G}(G)$ is planar, we also prove that the spectrum of group $G$ is contained in the Laplacian spectrum of graph $\mathcal{G}(G)$.