Given a group G, we define the power graph P(G) as follows: the vertices are the elements of G and two vertices x and y are joined by an edge if ⟨x⟩ ⊆ ⟨y⟩ or ⟨y⟩ ⊆ ⟨x⟩. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. We consider κ(G), the number of spanning trees of the power graph associated with a finite group G. In this paper, for a finite group G, first we represent some properties of P(G), then we are going to find some divisors of κ(G), and finally we prove that the simple group A6 ∼= L2(9) is uniquely determined by tree-number of its power graph among all finite simple groups.
