Connectivity of generating graphs of nilpotent groups

Let $G$ be $2$-generated group. The generating graph of $\mathrm{\Gamma }(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=⟨g,h⟩$. 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 $\mathrm{\Gamma }(G)$ in the case where $G$ is a finite nilpotent group. For example, we prove that if $G$ is nilpotent, then the graph obtained from $\mathrm{\Gamma }(G)$ by removing its isolated vertices is maximally connected and, if $\mathrm{\mid }G\mathrm{\mid }\ge 3$, also Hamiltonian. We pose several questions.