论文标题
在与有限组相关的电源图的树上
On the tree-number of the power graph associated with a finite groups
论文作者
论文摘要
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 $。显然，任何组的功率图总是连接的，因为组的身份元素与所有其他顶点相邻。我们考虑$κ（g）$，这是与有限组$ g $相关的动力图的跨树木数量。在本文中，对于有限的$ g $，首先，我们代表$ \ Mathcal {p}（g）$的一些属性，然后我们将找到$κ（g）$的一些除数，最后我们证明了简单的$ a_6 \ cong l_2（9）$由其在所有有限的有限的小组中唯一地确定。
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 $κ(G)$, 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 $κ(G)$, 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.
