论文标题
nilpotent群的非交通，非生成图
The non-commuting, non-generating graph of a nilpotent group
论文作者
论文摘要
对于nilpotent $ g $，令$ξ（g）$是$ g $的生成图的补充与$ g $的通勤图之间的区别，其顶点对应于$ g $的中央元素。也就是说，$ξ（g）$具有顶点set $ g \ setminus z（g）$，当且仅当它们不上下班并且不产生$ g $时，有两个顶点相邻。此外，令$ξ^+（g）$为$ξ（g）$的子图，其非分离顶点引起的。我们表明，如果$ξ（g）$具有边缘，则$ξ^+（g）$与直径$ 2 $或$ 3 $连接，而直径$ 3 $ case中的$ξ（g）=ξ^+（g）$。在无限的情况下，我们的结果更普遍地适用于所有最大亚组正常的组。当$ g $有限时，我们会更详细地探索$ g $和$ξ（g）$的结构之间的关系。
For a nilpotent group $G$, let $Ξ(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $Ξ(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $Ξ^+(G)$ be the subgraph of $Ξ(G)$ induced by its non-isolated vertices. We show that if $Ξ(G)$ has an edge, then $Ξ^+(G)$ is connected with diameter $2$ or $3$, with $Ξ(G) = Ξ^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $Ξ(G)$ in more detail.
