The Non-Commuting, Non-Generating Graph of a Nilpotent Group

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