Generalized Gardiner–Praeger graphs and their symmetries

• Published in: Discrete mathematics Vol. 344; no. 3; p. 112263
• Main Authors: Miklavič, Štefko, Šparl, Primož, Wilson, Stephen E.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2021
• Subjects: [formula omitted] graph; Arc-transitive; Automorphism; Half-arc-transitive; Symmetry; Tetravalent; Automorphism; Tetravalent; Arc-transitive; Half-arc-transitive; CPM graph; Symmetry
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2020.112263

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.