Adjacency spectrum of power graphs of $\mathbb{Z}_m \times \mathbb{Z}_n$

Abstract

The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. In this paper, first, we obtain the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any two positive integers $m$ and $n$. Then, we determine the full spectrum of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for some particular values of $m$ and $n$. Received: 20 January 2025 | Accepted: 13 January 2026