@article {
author = {Rahbariyan, Sakineh},
title = {On the tree-number of the power graph associated with some finite groups},
journal = {AUT Journal of Mathematics and Computing},
volume = {5},
number = {2},
pages = {81-89},
year = {2024},
publisher = {Amirkabir University of Technology},
issn = {2783-2449},
eissn = {2783-2287},
doi = {10.22060/ajmc.2023.21910.1123},
abstract = {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.},
keywords = {power graph,tree-number,simple group},
url = {https://ajmc.aut.ac.ir/article_5064.html},
eprint = {https://ajmc.aut.ac.ir/article_5064_0b4ed94fe78ea85cead87b6cd03b6c30.pdf}
}