TY - JOUR

T1 - Generalized Gardiner–Praeger graphs and their symmetries

AU - Miklavič, Štefko

AU - Šparl, Primož

AU - Wilson, Stephen E.

N1 - Funding Information:
All authors acknowledge support by the Slovenian Research Agency bilateral research project BI-US/18-20-075 .
Funding Information:
P. Šparl acknowledges support by the Slovenian Research Agency (research core funding No. P1-0285 and research projects J1-9108 , J1-9110 , J1-1694 , J1-1695 ).
Funding Information:
Š. Miklavič acknowledges support by the Slovenian Research Agency (research core funding No. P1-0285 and research projects N1-0062 , J1-9110 , J1-1695 , N1-0140 ).
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2021/3

Y1 - 2021/3

N2 - 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.

AB - 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.

KW - Arc-transitive

KW - Automorphism

KW - CPM graph

KW - Half-arc-transitive

KW - Symmetry

KW - Tetravalent

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U2 - 10.1016/j.disc.2020.112263

DO - 10.1016/j.disc.2020.112263

M3 - Article

AN - SCOPUS:85098851671

VL - 344

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

M1 - 112263

ER -