﻿ 关于弧传递Cayley图的判定 On Criterion of Arc-Transitive Cayley Graphs

Pure Mathematics
Vol.07 No.04(2017), Article ID:21346,5 pages
10.12677/PM.2017.74036

On Criterion of Arc-Transitive Cayley Graphs

Xue Yu

School of Mathematics and Statistics, Yunnan University, Kunming Yunnan

Received: Jun. 22nd, 2017; accepted: Jul. 7th, 2017; published: Jul. 13th, 2017

ABSTRACT

In 1938, R. Fruchet proved that for any given abstract group, there is a graph of it as an automorphism group. Since then, this area, which is about using the groups to study the graphs, opened the curtain. However, extensive research in this area began in 1960, especially in the last 30 years, where a number of important tasks were done. In this paper, we study a branch of graph theory that is, Cayley graph and its decision, especially the arc transitive graph. Firstly, by studying the properties of graph and the exchange of group, we get the main theorem of this paper. Secondly, according to the definition of the normal arc transitive graph and the pushing process of the main conclusion, a judgment condition of the normal arc transitive graph is given.

Keywords:Vertex-Transitive Graph, Arc-Transitive Graph, Cayley Graph, Orbital Graph

1938年R. Fruchet证明了对于任意给定的抽象群，都存在一个图以它为自同构群。自此，关于利用群来研究图这一领域，揭开了帷幕。但是，这个领域的广泛研究则是从1960年才真正开始的，尤其是最近30年，在这方面完成了很多重要的工作。本文主要研究了图论的一个分支，即：Cayley图以及它的判定，尤其是弧传递Cayley图的判定。首先，通过研究图的性质以及群的交换性，从而得出本文的主要定理。其次，根据正规弧传递Cayley图的定义以及主要结论的推导过程，得出了一个关于正规弧传递Cayley图的判定条件。

Copyright © 2017 by author and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

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1. 引言

1) 若的X-边传递Cayley图，则是弧传递图。

2) 若是交换群，则是X-弧传递图；若是非交换群，则

2. 预备知识

1)的弧传递有向图。

2)的弧传递图当且仅当

3)的边传递图。

4)不是的弧传递图当且仅当

1) 因为，所以，则有。又，故可以用中元素射到的任意一条弧，故的弧传递有向图。

2) 由1)知：的弧传递有向图。又为无向图当且仅当，所以的弧传递图当且仅当

3) 设，则有，所以的边集是，故，从而的边传递图。

4) 因为，所以，因此不是的弧传递图。反之亦然。

1) 若为G-弧传递图，则为自对偶Orbital图。

2) 若为G-边传递图，则为广义的Orbital图。

2) 因为为G-边传递图，且上是传递的，则上恰有两个轨道，则，那么为无向图，所以上的传递置换群的一个广义的Orbital图。

3. 定理的证明

2) 若是交换群，则的一个同构映射，即：。所以，那么是X-弧传递图。现在设是非交换群，则

。由 [3] Frattini论断知，。设如定理1.1证明中所定义的，则有-弧传递图。证毕。

2) 在推论1.2中，“”不一定成立。

On Criterion of Arc-Transitive Cayley Graphs[J]. 理论数学, 2017, 07(04): 277-281. http://dx.doi.org/10.12677/PM.2017.74036

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