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

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.

13 pages