Connectivity of generating graphs of nilpotent groups

Abstract

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.

Connected Components in the Invariably Generating Graph of a Finite Group

“…Indeed, in a finite nilpotent group, every maximal subgroup is normal; therefore, the concepts of generation and invariable generation coincide. See Harper and Lucchini [24] for results on the generating graph of finite nilpotent groups.…”

Forbidden subgraphs in generating graphs of finite groups

“…An open question is whether the subgraph ∆(G) of Γ(G) induced by the non-isolated vertices is connected, for every finite group G. The answer is positive if G is soluble [7] and in this case the diameter of ∆(G) is at most three [16]. In [14] it is proved that Andrea Lucchini & Daniele Nemmi when G is nilpotent, then ∆(G) is maximally connected, i.e. the connectivity of the graph ∆(G) equals its minimum degree (recall that the connectivity of a finite graph Γ is the least size of a subset X of the set V (Γ) of the vertices such that the induced subgraph on V (Γ) X is disconnected).…”

Forbidden subgraphs in generating graphs of finite groups

“…An open question is whether the subgraph ∆(G) of Γ(G) induced by the non-isolated vertices is connected, for every finite group G. The answer is positive if G is soluble [7] and in this case the diameter of ∆(G) is at most three [15]. In [13] it is proved that when G is nilpotent, then ∆(G) is maximally connected.…”