On the Cayleyness of Praeger-Xu graphs

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 t...

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Bibliographic Details
Published inJournal of combinatorial theory. Series B Vol. 152; pp. 55 - 79
Main Authors Jajcay, R., Potočnik, P., Wilson, S.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2022
Subjects
Automorphism group
Cayley graph
Praeger-Xu graph
Tetravalent graph
Vertex-transitive graph
Praeger-Xu graph
Automorphism group
Vertex-transitive graph
Cayley graph
Tetravalent graph
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Summary: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.
ISSN:0095-8956
1096-0902
DOI:10.1016/j.jctb.2021.09.003