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.

中文翻译：

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广义 Gardiner-Praeger 图及其对称性

如果图的自同构群的子群在图的顶点集和边集上可传递地作用，但在图的弧集上不传递，则它在图上作用 {\em half-arc-transitively} . 如果图的完全自同构群作用为半弧传递，则称该图为 {\em half-arc-transitive}。1994 年 Gardiner 和 Praeger 介绍了两个四价弧传递图族，称为 $C^{\pm 1}$ 和 $C^{\pm \varepsilon}$ 图，它们在表征四价图允许自同构的弧传递群与正规的初等阿贝尔子群，使得相应的商图是一个圈。所有 Gardiner-Praeger 图都是弧传递的，但承认半弧传递的自同构群。最近，Poto\v cnik 和 Wilson 介绍了 CPM 图系列，它们是 Gardiner-Praeger 图的推广。大多数这些图是弧传递的，但其中一些是半弧传递的。事实上，至少在 $1000$ 之前，每个具有大于 $2$ 阶的顶点稳定器的奇数半径的四价半弧传递松散连接图与 CPM 图同构。在本文中，我们确定 CPM 图的自同构群并研究它们之间的同构。此外，我们确定这些图中哪些是 $2$-arc-transitive，哪些是 arc-transitive 但不是 $2$-arc-transitive，以及哪些是 half-arc-transitive。至少在 $1000$ 之前，每个具有大于 $2$ 阶的顶点稳定器的奇数半径的四价半弧传递松散连接图与 CPM 图同构。在本文中，我们确定 CPM 图的自同构群并研究它们之间的同构。此外，我们确定这些图中哪些是 $2$-arc-transitive，哪些是 arc-transitive 但不是 $2$-arc-transitive，以及哪些是 half-arc-transitive。至少在 $1000$ 之前，每个具有大于 $2$ 阶的顶点稳定器的奇数半径的四价半弧传递松散连接图与 CPM 图同构。在本文中，我们确定 CPM 图的自同构群并研究它们之间的同构。此外，我们确定这些图中哪些是 $2$-arc-transitive，哪些是 arc-transitive 但不是 $2$-arc-transitive，以及哪些是 half-arc-transitive。