## Abstract

This paper discusses a family of graphs, called Praeger-Xu graphs and denoted PX(n,k) here, introduced by C.E. Praeger and M.-Y. Xu in 1989. These tetravalent graphs are distinguished by having large symmetry groups; their vertex-stabilizers can be arbitrarily larger than the number of vertices in the graph. This paper does the following: (1) exhibits a connection between vertex-transitive groups of symmetries in a Praeger-Xu graph and certain linear codes, (2) characterizes those linear codes, (3) characterizes Praeger-Xu graphs PX(n,k) which are Cayley, (4) shows that every PX(n,k) is quasi-Cayley, and (5) constructs an infinite family of Praeger-Xu graphs in which a smallest vertex-transitive group of symmetries has arbitrarily large vertex-stabiliser.

Original language | English (US) |
---|---|

Pages (from-to) | 55-79 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 152 |

DOIs | |

State | Published - Jan 2022 |

## Keywords

- Automorphism group
- Cayley graph
- Praeger-Xu graph
- Tetravalent graph
- Vertex-transitive graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics