一些给定阶数的七度对称图

发布时间：2019-07-05 05:22

【摘要】：一个图的图自同构群作用在它的弧集上是传递的,则称这个图为对称的.本文研究群论在图论中的应用,其对象是具有某种对称性的图,主要方法是通过图的自同构群来研究图的对称性.本文主要工作是分类和计数几类具有给定阶数的七度对称图.第一章,首先给出有关对称图的研究背景及意义,其次简要介绍与对称图相关的概念及引理.第二章,给出一个二倍的素数平方阶的7度对称图的分类.证明了阶数为8的7度对称图只有一个是完全图K8,阶数大于8的有四个PSU(3,5)的陪集图和一个素数平方阶的特殊交换Cayley有向图的标准双覆盖.第三章,给出了所有的2pq阶的7度连通对称图,其中的p,q是互不相同的素数.当q = 2时,4p阶7度连通对称图有且仅有一个,为K8,对于奇素数p和q,具有可解自同构群的7度连通1-正则图是一个无限类,还有4个零散的具有不可解自同构群的图,它们分别是1,2,3-弧传递的.特别地,这四个零散的图其中一个是本原的,还有两个是双本原的.第四章,给出了所有的4pq阶的7度连通对称图,其中的p,q是互不相同的素数.当q = 2时,有且仅有两个8p阶的7度连通对称图,对于奇素数p和q,有8个零散的具有不可解自同构群的1-传递7度连通图.特别地,这八个零散的图中有两个是78阶7度对称图的标准双覆盖.
[Abstract]:If the automorphism group of a graph acts on its arc set, it is called symmetric. In this paper, the application of group theory in graph theory is studied. the object of group theory is graph with some symmetry. the main method is to study the symmetry of graph by automorphism group of graph. The main work of this paper is to classify and count several kinds of seven-degree symmetric graphs with a given order. In the first chapter, the research background and significance of symmetric graph are given, and then the concepts and Lemma related to symmetric graph are briefly introduced. In chapter 2, we give a classification of 7-degree symmetric graphs with twice the square order of prime numbers. It is proved that only one symmetric graph of order 8 is a complete graph K8. if the order is greater than 8, there are four coset graphs of PSU (3, 5) and a standard double covering of a special commutative Cayley directed graph of square order of prime number. In chapter 3, we give all the 7-degree connected symmetric graphs of order 2pq, where p, Q is a different prime number. When Q = 2, there is only one connected symmetric graph of order 4p, which is K8. for odd prime numbers p and Q, the 7-degree connected 1-regular graph with solvable automorphism group is an infinite class, and there are four scattered graphs with unsolvable automorphism group, which are 1, 2, 3-arc transitive respectively. In particular, one of the four scattered graphs is primitive and the other two are biprimitive. In chapter 4, we give all the 7-degree connected symmetric graphs of order 4pq, where p, Q is a different prime number. When Q = 2, there are and only two 7-degree connected symmetric graphs of order 8p. For odd prime numbers p and Q, there are eight scattered 1-transitive 7-degree connected graphs with unsolvable automorphism groups. In particular, two of these eight scattered graphs are standard double covers of 7-degree symmetric graphs of order 78.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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