A B S T R A C T This paper discusses a family of graphs, called PraegerXu 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 PraegerXu 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. (c) 2021 Elsevier Inc. All rights reserved.
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