On the connectivity of the non-generating graph

TL;DR

This paper studies the connectivity properties of a graph derived from a finite group, showing conditions under which it is connected and classifying groups where it is disconnected.

Contribution

It introduces the graph (G) derived from the non-generating graph of a finite group and characterizes its connectivity based on the group's derived subgroup.

Findings

If the derived subgroup is not nilpotent, (G) is connected with diameter at most 5.

Complete classification of finite groups where (G) is disconnected.

Provides structural insights into the non-generating graph of finite groups.

Abstract

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained from the non-generating graph of $G$ by deleting the universal vertices. We prove that if the derived subgroup of $G$ is not nilpotent, then this graph is connected, with diameter at most 5. Moreover we give a complete classification of the finite groups $G$ such that $\Sigma(G)$ is disconnected.