广义Gardiner-Praeger图及其对称性
Generalized Gardiner-Praeger graphs and their symmetries
AI总结：
本文研究CPM图（Gardiner-Praeger图的推广），确定其自同构群、同构关系，并区分2-弧传递、弧传递非2-弧传递及半弧传递的CPM图类型。
摘要：
图自同构群的一个子群若在图的顶点集和边集上传递作用，但在弧集上不传递作用，则称其在该图上‘半弧传递’作用。若图的全自同构群半弧传递作用，则该图称为‘半弧传递图’。1994年，Gardiner和Praeger引入了两类四价弧传递图，称为$C^{\pm 1}$和$C^{\pm \varepsilon}$图，它们在刻画具有正规初等交换子群且对应商图为圈的四价弧传递图中发挥重要作用。所有Gardiner-Praeger图都是弧传递的，但允许半弧传递的自同构群。最近，Poto\v cnik和Wilson引入了CPM图族，它是Gardiner-Praeger图的推广。大多数CPM图是弧传递的，但部分是半弧传递的。事实上，至少在阶数1000以内，每个顶点稳定子群阶大于2的奇数半径四价半弧传递松散连接图都与某个CPM图同构。本文确定了CPM图的自同构群，并研究了它们之间的同构关系；此外，还确定了哪些CPM图是2-弧传递的，哪些是弧传递但非2-弧传递的，以及哪些是半弧传递的。
英文摘要：
A subgroup of the automorphism group of a graph acts {\em half-arc-transitively} on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be {\em half-arc-transitive}. In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the $C^{\pm 1}$ and the $C^{\pm \varepsilon}$ graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner-Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Poto\v cnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner-Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order $1000$, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than $2$ is isomorphic to a CPM graph. In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are $2$-arc-transitive, which are arc-transitive but not $2$-arc-transitive, and which are half-arc-transitive.
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