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arXiv 1409.8080math.GRmath.CO
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弧传递图的阶

On the orders of arc-transitive graphs

Marston D. E. Conder, Cai-Heng Li, Primoz Potocnik

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摘要：

一个图称为{ \em弧传递}（或{ \em对称}），如果它的自同构群在邻顶点的序偶上有一个轨道，并且2-弧传递它的自同构群在长度为2的有序路上有一个轨道。在本文中，我们考虑这样的图的阶，对于给定的价。证明了对任意给定的正整数$k$，对于某个素数$p$，只存在1000个连通的3-度2-弧传递图，其阶为$kp$;如果是$d\ge 4$，则对于某个素数$p$，只存在1000个连通的$d$ -度2-弧传递图，其阶为$kp$或$kp^2$.我们还证明了存在无穷多个（偶数）$k$的值，使得只有无穷多个连通的$kp$阶3-度对称图，其中$p$是素数.

英文摘要：

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $kp^2$ for some prime $p$. We also prove that there are infinitely many (even) values of $k$ for which there are only finitely many connected 3-valent symmetric graphs of order $kp$ where $p$ is prime.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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