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 language | English (US) |
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Pages (from-to) | 581-589 |
Number of pages | 9 |
Journal | Ars Mathematica Contemporanea |
Volume | 17 |
Issue number | 2 |
DOIs | |
State | Published - 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