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 ${\tilde{\Gamma }}_{\mathfrak{F}}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $⟨g,h⟩\notin \mathfrak{F}$. Moreover we denote by ${\mathcal{I}}_{\mathfrak{F}}(G)$ the set of the isolated vertices of ${\tilde{\Gamma }}_{\mathfrak{F}}(G).$ We address the following question: to what extent the fact that ${\mathcal{I}}_{\mathfrak{F}}(G)$ is a subgroup of $H$ for any $H\le G,$ implies that the graph ${\Gamma }_{\mathfrak{F}}G)$ obtained from ${\tilde{\Gamma }}_{\mathfrak{F}}(G)$ by deleting the isolated vertices is a connected graph?