  Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the
graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$
are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial
structure of the distribution of generating pairs across $G$. In this paper we
study several natural graph theoretic properties related to the connectedness
of $\Gamma(G)$ in the case where $G$ is a finite nilpotent group. For example,
we prove that if $G$ is nilpotent, then the graph obtained from $\Gamma(G)$ by
removing its isolated vertices is maximally connected and, if $|G| \geq 3$,
also Hamiltonian. We pose several questions.
