Spectrum of proper power graphs of the direct product of certain finite groups

The proper power graph P* (G) of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups G(1) and G(2), where at least one of G(1) and G(2) is an EPO group. Also we provide the full spectrum and hence the energy of P* (G(1) x G(2)) whenever both G(1) and G(2) are EPO groups.