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

TL;DR

This paper investigates the properties of a graph constructed from a finite group based on a class of groups, focusing on how the structure of isolated vertices influences the graph's connectivity and regularity.

Contribution

It introduces the concept of the non-$rak F$ graph for finite groups and explores the relationship between isolated vertices forming subgroups and the graph's connectivity.

Findings

The non-$rak F$ graph exhibits semiregularity under certain conditions.

The structure of isolated vertices as subgroups affects the connectivity of the reduced graph.

Conditions are identified under which the graph remains connected despite isolated vertices.

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?