Generalized Gardiner–Praeger graphs and their symmetries

Štefko Miklavič, Primož Šparl, Stephen E. Wilson

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive. In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the C±1 and the C±ε graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner–Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Potočnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner–Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a CPM graph. In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.

Original languageEnglish (US)
Article number112263
JournalDiscrete Mathematics
Issue number3
StatePublished - Mar 2021


  • Arc-transitive
  • Automorphism
  • CPM graph
  • Half-arc-transitive
  • Symmetry
  • Tetravalent

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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