The complexity of power graph associated with finite p-groups

Given a finite group G , the power graph P ( G ) is the graph with a vertex set consisting of the elements of G , where two vertices x and y are joined by an edge if and only if 〈 x 〉 ⊆ 〈 y 〉 or 〈 y 〉 ⊆ 〈 x 〉 . We consider κ ( G ) , the number of spanning trees of P ( G ) , called the complexity of P ( G ) . The purpose of this paper is twofold. First, we derive some explicit formulas concerning the complexity κ ( G ) for a finite p -group G . Second, we show that the simple group A 5 is uniquely determined by the number of spanning trees of its power graph among all finite groups.