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

TL;DR

This paper computes the characteristic polynomial of the power graph of the direct product of two finite cyclic groups, providing explicit formulas and spectra for specific cases, advancing understanding of algebraic graph spectra.

Contribution

It determines the characteristic polynomial of power graphs for the direct product of any two finite cyclic groups, including simplifications and spectra for particular cases.

Findings

Explicit characteristic polynomial formulas for $ ext{Power}(Z_m imes Z_n)$

Simplified spectra for specific $(m,n)$ cases

Enhanced understanding of algebraic graph spectra for cyclic groups

Abstract

The power graph $\mathscr{P}(G)$ 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 $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ 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.