  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.
