# 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.

## Key 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 $\mathcal{G}(G)$.