Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24495220240401On the tree-number of the power graph associated with some finite groups8189506410.22060/ajmc.2023.21910.1123ENSakinehRahbariyanDepartment of Mathematics, Faculty of Sciences, Arak University, Arak, IranJournal Article20221104Given 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.https://ajmc.aut.ac.ir/article_5064_0b4ed94fe78ea85cead87b6cd03b6c30.pdf