Semiregularity and connectivity of the non-$\mathfrak F$ graph of a finite group

Abstract

Given a class $\mathfrak F$ of finite groups, we consider the graph $\widetilde\Gamma_{\mathfrak F}(G)$ 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 $\widetilde\Gamma_{\mathfrak F}(G).$ 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 $\Gamma_{\mathfrak F}G)$ obtained from $\widetilde\Gamma_{\mathfrak F}(G)$ by deleting the isolated vertices is a connected graph?