Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups

The power graph P(G) of a group G is defined as the simple graph with vertex set G, and where two distinct vertices x and y are joined by an edge if and only if either x= yᵏ or y= xᵏ, k ∈ N. Here we determine the characteristic polynomial of P(Zₘ × Zₙ) for any positive integers m and n. Additionally, for some particular values of m and n, we simplify the above characteristic polynomials and provide the full spectrum in a few cases.