## Abstract

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 language | English (US) |
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Article number | 112263 |

Journal | Discrete Mathematics |

Volume | 344 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2021 |

## Keywords

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

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics