Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups

Abstract

The power graph $\mathcal{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x=y^k$ or $y=x^k$, $k\in \mathbb{N}$. Here we determine the characteristic polynomial of $\mathcal{P}({\mathbb{Z}}_m\times {\mathbb{Z}}_n)$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases.