四度点传递双循环图

发布时间：2018-03-05 01:14

本文选题：双Cayley图　切入点：点传递图　出处：《北京交通大学》2017年硕士论文　论文类型：学位论文

【摘要】：一个图r称为是群H上的双Cayley图,如果r的全自同构群中有一个同构于H的子群,使得这个子群在r的顶点集合上作用半正则且恰好有两条轨道.若H为循环群或交换群,则相应地称图r为双循环图或双交换图.本文将重点关注具有特定对称性质的双循环图.一个图称为是边传递(点传递或弧传递的),如果它的全自同构群在其边集合(点集合或弧集合)上作用传递.目前,人们已给出了三度边传递双交换图的分类(特别地,所有三度边传递双循环图也是已知的).最近,Kovacs等人分别给出了四度、五度边传递双循环图的完全分类.受这些工作的启发,本文研究了四度点传递双循环图.设n为奇数,本文给出了 2n阶的四度连通点传递非Cayley的双循环图的完全分类.作为副产品,构造出了一个新的点传递非Cayley图的无限类.作为上述结果的应用,本文还给出2p2(p为素数)阶四度点传递非Cayley图的完全分类.本文是如下组织的:第一章介绍了本文所用到的一些有关群与图的基本概念,并综述了与本文研究内容相关的研究问题以及背景知识.第二章介绍了有关双Cayley图的一些基本结果,并构造了一类四度连通点传递非Cayley的双循环图.第三章给出了奇数阶循环群上的四度连通点传递非(Cayley的双Cayley图的完全分类.第四章给出了四度连通二倍素数平方阶点传递非Cayley图的完全分类.
[Abstract]:A graph r is called a double Cayley graph on a group H. if a subgroup of the total automorphism group of r has a subgroup isomorphic to H such that the subgroup is semi-regular and has exactly two orbits on the vertex set of r, if H is a cyclic group or a commutative group, Then the graph r is called a double cyclic graph or a double commutative graph. In this paper, we will focus on the bicyclic graph with certain symmetry. A graph is called edge transitive (vertex transitive or arc transitive) if its total automorphism group is on its edge. Action transfer on a set (point set or arc set). Currently, The classification of three degree edge transitive double commutative graphs has been given (in particular, all three degree edge transitive double circulant graphs are also known. Recently, Kovacs et al. Has given a complete classification of three degree and five degree edge transitive double circulant graphs. In this paper, we study four degree vertex transitive bicyclic graphs. Let n be odd. In this paper, we give the complete classification of 4-degree connected vertex-transitive non-#en0# bicyclic graphs of order 2n. As a by-product, A new infinite class of vertex-transitive non-#en0# graphs is constructed as an application of the above results. In this paper, we also give a complete classification of the non-#en0# graphs of the fourth degree point transitive in the order 2p2p (prime number). This paper is organized as follows: in Chapter 1, some basic concepts about groups and graphs used in this paper are introduced. In the second chapter, some basic results of double Cayley graphs are introduced. In chapter 3, we give a complete classification of the four-degree connected vertex-transitive non-#en1# graphs on the odd order cyclic group. Chapter 4th gives the fourth degree connected diploid prime square. The complete classification of order transitive non-Cayley graphs.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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