Power graphs via their characteristic polynomial

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 χ(x,λ) = 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 p²q, where p, q are distinct prime numbers.