有限群的非$\mathfrak F$图的半正则性和连通性
Semiregularity and connectivity of the non-$\mathfrak F$ graph of a finite group
摘要：
给定一个有限群类$\mathfrak F$，我们考虑图$\widetildeΓ_{\mathfrak F}(G)$，其顶点是$G$的元素，并且其中两个顶点$g,h\in G$相邻当且仅当$\langle g,h\rangle\notin\mathfrak F$ .此外，我们表示$\mathcal{I}_{\mathfrak F}(G)$的孤立顶点的集合$\widetildeΓ_{\mathfrak F}(G).$我们解决以下问题：在何种程度上的事实，$\mathcal{I}_{\mathfrak F}(G)$是一个子群的$H$的任何$H\leq G,$意味着图$Γ_{\mathfrak F}G)$从$\widetildeΓ_{\mathfrak F}(G)$删除孤立顶点是一个连通图？
英文摘要：
Given a class $\mathfrak F$ of finite groups, we consider the graph $\widetildeΓ_{\mathfrak F}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h\rangle\notin\mathfrak F$. Moreover we denote by $\mathcal{I}_{\mathfrak F}(G)$ the set of the isolated vertices of $\widetildeΓ_{\mathfrak F}(G).$ We address the following question: to what extent the fact that $\mathcal{I}_{\mathfrak F}(G)$ is a subgroup of $H$ for any $H\leq G,$ implies that the graph $Γ_{\mathfrak F}G)$ obtained from $\widetildeΓ_{\mathfrak F}(G)$ by deleting the isolated vertices is a connected graph?
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