  Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the
vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an
edge if $\langle x \rangle \subseteq \langle y \rangle$ or $\langle y \rangle
\subseteq \langle x \rangle$. 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 $\kappa(G)$, 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 $\mathcal{P}(G)$, then we are going
to find some divisors of $\kappa(G)$, and finally we prove that the simple
group $A_6\cong L_2(9)$ is uniquely determined by tree-number of its power
graph among all finite simple groups.
