%0 Journal Article
%T On the tree-number of the power graph associated with some finite groups
%J AUT Journal of Mathematics and Computing
%I Amirkabir University of Technology
%Z 2783-2449
%A Rahbariyan, Sakineh
%D 2024
%\ 04/01/2024
%V 5
%N 2
%P 81-89
%! On the tree-number of the power graph associated with some finite groups
%K power graph
%K tree-number
%K simple group
%R 10.22060/ajmc.2023.21910.1123
%X 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.
%U https://ajmc.aut.ac.ir/article_5064_0b4ed94fe78ea85cead87b6cd03b6c30.pdf