Connectivity of generating graphs of nilpotent groups

Scott Harper, Andrea Lucchini

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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.
Original languageEnglish
Pages (from-to)1183-1195
Number of pages13
JournalAlgebraic Combinatorics
Volume3
Issue number5
DOIs
Publication statusPublished - 12 Oct 2020

Keywords

  • Generating graph
  • connectivity
  • nilpotent groups

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