本文選題：1-正則圖　切入點(diǎn)：凱萊圖　出處：《北京交通大學(xué)》2017年碩士論文

【摘要】：凱萊圖是圖的對稱性研究中的重要課題之一,因其構(gòu)造的簡潔性和高度的對稱性在數(shù)學(xué)及眾多應(yīng)用學(xué)科中發(fā)揮著重要的作用.令r為有限群G上的一個凱萊圖.如果正則子群R(G)在r的全自同構(gòu)群Aut(Γ)中正規(guī),則稱r為G上的正規(guī)凱萊圖;如果Aut(Γ)作用在r的弧集上正則,則稱r為1-正則圖.特別的,稱二面體群上的凱萊圖為二面體圖.本文主要研究二倍素數(shù)度1-正則的二面體圖.論文結(jié)構(gòu)組織如下.第1章緒論部分,主要介紹了本文所要用到的有限群論和圖論的基本概念,以及與1-正則的二面體圖有關(guān)的背景知識.第2章主要介紹了一些與二面體圖和字典式積圖有關(guān)的結(jié)果.第3章給出了點(diǎn)穩(wěn)定子群為二面體群的正規(guī)的1-正則二面體圖的完全分類.結(jié)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]中的結(jié)果,二倍素數(shù)度正規(guī)的1-正則二面體圖被決定.第4章給出了六度非正規(guī)的1-正則二面體圖的刻畫.令r表示一個連通的六度非正規(guī)的1-正則二面體圖.本章證明了 Aut(r)總存在一個半正則循環(huán)子群H,使得商圖ΓH同構(gòu)于12階完全二部圖K6,6,4階完全圖K4,長為4或6的圈,或者字典式積圖K4[2K1],并且給出了 Aut(Γ)的刻畫.第5章為結(jié)束語部分.
[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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O157.5

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