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On the Cayleyness of Praeger-Xu graphs
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2021-09-29 , DOI: 10.1016/j.jctb.2021.09.003
R. Jajcay , P. Potočnik , S. Wilson

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.



中文翻译:

关于 Praeger-Xu 图的凯莱性

本文讨论了一系列图,称为 Praeger-Xu 图,表示为 PX(n,)在这里,由 CE Praeger 和 M.-Y 介绍。Xu 在 1989 年。这些四价图的特点是具有大的对称群;它们的顶点稳定器可以任意大于图中的顶点数。本文做了以下工作:(1) 展示了 Praeger-Xu 图中的顶点传递对称群与某些线性代码之间的联系,(2) 表征这些线性代码,(3) 表征 Praeger-Xu 图PX(n,) 是 Cayley,(4) 表明,每 PX(n,) 是拟凯莱,并且 (5) 构造了一个无限的 Praeger-Xu 图族,其中最小的顶点传递对称群具有任意大的顶点稳定子。

更新日期:2021-09-29
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