几类弧传递图的研究

发布时间：2018-08-18 20:22

【摘要】：本文主要研究代数图论中与弧传递图相关的两个问题：一个是刻画包含弧正则子群的2-弧传递图,另一个是限定某些条件的5度弧传递图.首先,在第三和第四章中,我们对包含弧正则子群G的(X,2)-弧传递图Γ,当G(?)X,X在VΓ上作用拟本原的情形,给了一般性的刻画.证明这样的图只能是PA型,AS型或者TW型的.并且对其中的AS型和PA型分别构造了两个无限类的例子以及一些零散的例子.特别地,我们指出PA型的例子是难以构造的,它的存在性问题自Praeger在1992年提出,直到2006年才被李才恒和A. Seress在[Constructions of quasiprimitive two-arc transitive graphs of product action type, Finite Geometries, Groups and Computation (2006),115-124]中解决.在接下来的第五到第七章中,我们主要对5度1-传递和2-传递Cayley图进行了研究.在第五章中,我们通过分类所有无核的5度1-传递Cayley图,证明了周进鑫和冯衍全2010年在[On symmetric graphs of valency five, Discrete Math.310 (2010),1725-1732.]上的一个主要结论,即：所有非交换单群上的5度1-传递Cayley图都是正规的.此外,在研究无核的5度弧传递Cayley图的过程中,我们找到了一个2-弧传递的Cayley图Cay(G,S),满足Aut(G,S)在S上传递但非2-传递.从而回答了李才恒2008年在[On automorphism groups of quasiprimitive 2-arc transitive graphs, J Algebr Comb 28 (2008),261-270.]上提出是否存在这样Cayley图的问题.在第六章中,我们对非交换单群上的5度2-传递Cayley图的正规性进行了研究.证明了：除了交错群A39,A59,A119之外的所有非交换单群上的5度2-传递Cayley图都是正规的.另外,还对非交换单群上具有可解点稳定子群的5度弧传递Cayley图的正规性进行了研究.证明了：除了交错群A39和A79之外的所有非交换单群上具有可解点稳定子群的5度弧传递Cayley图都是正规的.在第七章中,我们分别构造了交错群A39,A59,A79和A119上的5度弧传递Cayley图非正规的例子.在最后一章中,我们对无平方因子阶或者2倍无平方因子阶的5度弧传递图进行了分类.
[Abstract]:In this paper, two problems related to arc-transitive graphs in algebraic graph theory are studied: one is to characterize 2-arc-transitive graphs containing arc-regular subgroups, the other is a 5-degree arc-transitive graph with certain conditions. First, in the third and fourth chapters, we give a general characterization of the (X2) -arc transitive graph 螕 containing an arc regular subgroup G when G (?) XX acts on the quasi-primitive of V 螕. It is proved that such a graph can only be PA type as or TW type. Two infinite class examples and some scattered examples are constructed for as type and PA type respectively. In particular, we point out that the example of PA type is difficult to construct. Its existence problem was proposed by Praeger in 1992 and only solved by Li Caiheng and A. Seress in [Constructions of quasiprimitive two-arc transitive graphs of product action type, Finite Geometries, Groups and Computation (2006] 115-124 in 2006. In the following chapters 5 to 7, we study the 5-degree 1-transitive and 2-transitive Cayley diagrams. In Chapter 5, by classifying all seedless 5-degree 1-transitive Cayley diagrams, we prove that Zhou Jinxin and Feng Yanquan in [On symmetric graphs of valency five, Discrete Math.310 (2010) 1725-1732.] One of the main conclusions is that all 1-transitive Cayley graphs of degree 5 on nonabelian simple groups are normal. In addition, we find a 2-arc-transitive Cayley graph (Cay), which satisfies Aut (GCS) transfer on S, but not 2-transitive, in the process of studying the 5-degree arc-transitive Cayley diagram without nuclei. Thus, Li Caiheng answered Li Caiheng in [On automorphism groups of quasiprimitive 2-arc transitive graphs, J Algebr Comb 28 (2008) 261-270.] The question of whether there is such a Cayley graph is proposed on this paper. In chapter 6, we study the normality of 2-transitive Cayley graphs of degree 5 on nonabelian simple groups. It is proved that all 2-transitive Cayley graphs of degree 5 on nonabelian simple groups except the staggered group A39 ~ (9) A _ (59) ~ (9) A _ (119) are normal. In addition, the normality of arc-transitive Cayley graphs of degree 5 on nonabelian simple groups with solvable point stable subgroups is studied. It is proved that all arc-transitive Cayley graphs with solvable vertex-stable subgroups on all noncommutative simple groups except staggered groups A39 and A79 are normal. In the seventh chapter, we construct the examples of 5 degree arc-transitive Cayley graphs on the staggered groups A39, A59, A79 and A119, respectively. In the last chapter, we classify the five degree arc transfer graph with square free factor order or double square factor order.
【学位授予单位】：云南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5

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