论文标题
局部有限的顶点 - 连通图和具有有限价值和有限边缘多重性的coset图
Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity
论文作者
论文摘要
众所周知，一个简单的$ G $ -ARC传输图可以表示为组$ g $的coset图。该表示形式扩展到$ g $ -ARC-arc-arc-transitive coset图$ \ cos（g，h，j）$，具有有限的货币和有限的边缘 - 多层性，其中$ h，j $分别是$ g $的顶点和事件边缘的稳定器。在适当的有限假设下，显示了一个$ g = g = g = g = g = g = r $，$ | z | = 2 $和$ | = 2 $和$ | = $ | $ g $ -rotary}地图如果$ | az | $是有限的，而如果$ | zz^a | $是有限的，则{\ it $ g $ -bi-rotary}地图。 $ g $ - 风向图可以表示为$ g $的固定几何形状，从而扩展了coset图的概念。但是，$ g $ -bi-Rotary地图没有这样的表示，并且除了面部和边缘之间的发生率外，还必须指定面部边界周期。我们还提供了一个固定的几何形状构造，构造了旗帜规范地图$（v，e，f）$。在所有这些结构中，我们证明面部边界周期是常规的周期，当给定的组忠实地在$ v \ cup f $上行动时，这是简单的周期。
It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H, J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=ła,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,ła\r,łz\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rotary} map if $|az|$ is finite, and a {\it $G$-bi-rotary} map if $|zz^a|$ is finite. The $G$-rotary map can be represented as a coset geometry for $G$, extending the notion of a coset graph. However the $G$-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. We also give a coset geometry construction of a flag-regular map $(V,E,F)$. In all of these constructions we prove that the face boundary cycles are regular cycles which are simple cycles precisely when the given group acts faithfully on $V\cup F$.
