The non-$\mathfrak F$-graph of a finite group

TL;DR

This paper investigates a graph constructed from a finite group based on a formation, focusing on properties of isolated vertices and connectivity after their removal.

Contribution

It introduces and analyzes a new graph associated with a formation on a finite group, exploring subgroup and connectivity properties.

Findings

Isolated vertices form a subgroup under certain conditions.

Removing isolated vertices often results in a connected subgraph.

The structure of the graph reveals insights into the group's formation properties.

Abstract

Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of $G$? Is the subgraph obtained by deleting the isolated vertices a connected graph?