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.
Rahbariyan, S. (2024). On the tree-number of the power graph associated with some finite groups. AUT Journal of Mathematics and Computing, 5(2), 81-89. doi: 10.22060/ajmc.2023.21910.1123
MLA
Sakineh Rahbariyan. "On the tree-number of the power graph associated with some finite groups". AUT Journal of Mathematics and Computing, 5, 2, 2024, 81-89. doi: 10.22060/ajmc.2023.21910.1123
HARVARD
Rahbariyan, S. (2024). 'On the tree-number of the power graph associated with some finite groups', AUT Journal of Mathematics and Computing, 5(2), pp. 81-89. doi: 10.22060/ajmc.2023.21910.1123
VANCOUVER
Rahbariyan, S. On the tree-number of the power graph associated with some finite groups. AUT Journal of Mathematics and Computing, 2024; 5(2): 81-89. doi: 10.22060/ajmc.2023.21910.1123
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