On the Cayleyness of Praeger-Xu graphs

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.