Let G be 2-generated group. The generating graph Γ(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 Γ(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 Γ(G) by removing its isolated vertices is maximally connected and, if |G|⩾3, also Hamiltonian. We pose several questions.
- Generating graph
- nilpotent groups