Spectral Properties of Power Graphs of Metacyclic Groups

Abstract

For a group $\Omega$, the associated power graph $P(\Omega)$ is defined as the graph whose vertices are the elements of $\Omega$, with two distinct vertices $u,v\in \Omega$ being adjacent if either $u=v^m$ or $v=u^n$ for some $m,n \in \mathbb{N}$. In this paper, we completely characterise the structure of the power graph associated with the class of metacyclic groups. Building on this structural description, we derive explicit expressions for the characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices. Moreover, we obtain lower and upper bounds for the spectral radii of the adjacency and signless Laplacian matrices.