On the tree-number of finite $p$-groups

Given a group $G$, the power graph $mathcal{P}(G)$ is the graph with vertex set of $G$ and two vertices $x$ and $y$ are joined by an edge if $langle x angle subseteq langle y angle$ or $langle y angle subseteq langle x angle$. We consider $kappa(G)$, the number of spanning trees of $mathcal{P}(G)$. In this paper, we focus on $kappa(G)$, for a finite $p$-group $G$.