The non-commuting, non-generating graph of a nilpotent group

TL;DR

This paper investigates the structure and connectivity properties of a graph constructed from a nilpotent group, revealing that under certain conditions, the graph's non-isolated part is highly connected with small diameter.

Contribution

It introduces the graph $\Xi(G)$ for nilpotent groups and characterizes its connectivity and diameter, extending results to infinite groups with normal maximal subgroups.

Findings

If $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter 2 or 3.

In the diameter 3 case, $\Xi(G) = \Xi^+(G)$.

Results apply to infinite groups with all maximal subgroups normal.

Abstract

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.