Power graphs via their characteristic polynomial

Abstract

A power graph is defined a graph that it's vertices are the elements of group and two vertices are adjacent if and only if one of them is a power of the other. Suppose $A(X)$ is the adjacency matrix of graph $X$. Then the polynomial $\chi(X,\lambda)=det(xI-A(X))$ is called as characteristic polynomial of $X$. In this paper, we compute the characteristic polynomial of all power graphs of order $p^2q$, where $p,q$ are distinct prime numbers.