二倍素数度1-正则二面体图
本文选题：1-正则图 切入点：凯莱图 出处：《北京交通大学》2017年硕士论文
【摘要】：凯莱图是图的对称性研究中的重要课题之一,因其构造的简洁性和高度的对称性在数学及众多应用学科中发挥着重要的作用.令r为有限群G上的一个凯莱图.如果正则子群R(G)在r的全自同构群Aut(Γ)中正规,则称r为G上的正规凯莱图;如果Aut(Γ)作用在r的弧集上正则,则称r为1-正则图.特别的,称二面体群上的凯莱图为二面体图.本文主要研究二倍素数度1-正则的二面体图.论文结构组织如下.第1章绪论部分,主要介绍了本文所要用到的有限群论和图论的基本概念,以及与1-正则的二面体图有关的背景知识.第2章主要介绍了一些与二面体图和字典式积图有关的结果.第3章给出了点稳定子群为二面体群的正规的1-正则二面体图的完全分类.结合 Kwak 等人在[Journal of Combinatorial Theory,Series B,98(2008)585-598]以及Wang 等人在[Acta Mathematica Siirica,Chinese Series,3(2006)669-678]中的结果,二倍素数度正规的1-正则二面体图被决定.第4章给出了六度非正规的1-正则二面体图的刻画.令r表示一个连通的六度非正规的1-正则二面体图.本章证明了 Aut(r)总存在一个半正则循环子群H,使得商图ΓH同构于12阶完全二部图K6,6,4阶完全图K4,长为4或6的圈,或者字典式积图K4[2K1],并且给出了 Aut(Γ)的刻画.第5章为结束语部分.
[Abstract]:Calais graph is one of the most important topics in the study of symmetry of graphs. Because of its conciseness and high symmetry, it plays an important role in mathematics and many applied disciplines. Let r be a Calais graph on finite group G. if the regular subgroup RG) is normal in the totally automorphism group Aut (螕) of r, Then r is called a normal Calais graph on G. if Aut (螕) acts on the arc set of r, then r is called a 1-regular graph. In this paper, we study the dihedral graph with dihedral degree 1-regular dihedral graph. The structure of the paper is as follows. The basic concepts of finite group theory and graph theory used in this paper are introduced. In chapter 2, we mainly introduce some results related to dihedral graphs and dictionary-type product graphs. In chapter 3, we give the normal 1-regular dihedral subgroups with vertex stable subgroups as dihedral groups. Complete classification of plane maps. Combined with the results of Kwak et al. [Journal of Combinatorial Series Bf98 / 2008 / 585-598] and Wang et al. [Acta Mathematica Siirica / Chinese Series / 2006 / 669-678]. In chapter 4, the characterization of 1-regular dihedron graph of degree 6 is given. Let r denote a connected 1-regular dihedron graph of degree 6. In this chapter, it is proved that. There is always a semiregular cyclic subgroup H, such that quotient graph 螕 H is isomorphic to a complete graph of order K _ 6 ~ (6) of order K _ 4 with a cycle of 4 or 6. Or dictionary product graph K4 [2K1], and the characterization of Aut (螕) is given. Chapter 5 is the concluding part.
【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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