Laplacian spectra of power graphs of certain prime-power Abelian groups

The power graph G(S) of a semigroup S is a simple undirected graph whose vertex set is S itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph G(Z(p)(r) x Z(p2)(s)) in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of G(Z(p)(r) x Z(p2)(s)). We finally calculate the Laplacian polynomial of the power graph g(Z(pm)(n)).