The Laplacian spectrum of power graphs of some finite abelian p-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 $\mathcal{G}(G)$.