Lobe, edge, and arc transitivity of graphs of connectivity

Jack E. Graver, Mark E. Watkins

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph Λ and a "code" assigned to each orbit of Aut(Λ), there exists a unique lobe-transitive graph Γ of connectivity 1 whose lobes are copies of Λ and is consistent with the given code at every vertex of Γ. These results lead to necessary and sufficient conditions for a graph of connectivity 1 to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose automorphism groups is, respectively, vertextransitive, primitive, regular, Cayley, and Frobenius had been previously characterized in the literature.

Original languageEnglish (US)
Pages (from-to)581-589
Number of pages9
JournalArs Mathematica Contemporanea
Volume17
Issue number2
DOIs
StatePublished - 2019

Keywords

  • Connectivity
  • Edge-transitive
  • Lobe
  • Lobe-transitive
  • Orbit

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Algebra and Number Theory
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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