幾類弧傳遞圖的研究
發(fā)布時(shí)間：2018-08-18 20:22
【摘要】：本文主要研究代數(shù)圖論中與弧傳遞圖相關(guān)的兩個(gè)問題：一個(gè)是刻畫包含弧正則子群的2-弧傳遞圖,另一個(gè)是限定某些條件的5度弧傳遞圖.首先,在第三和第四章中,我們對(duì)包含弧正則子群G的(X,2)-弧傳遞圖Γ,當(dāng)G(?)X,X在VΓ上作用擬本原的情形,給了一般性的刻畫.證明這樣的圖只能是PA型,AS型或者TW型的.并且對(duì)其中的AS型和PA型分別構(gòu)造了兩個(gè)無限類的例子以及一些零散的例子.特別地,我們指出PA型的例子是難以構(gòu)造的,它的存在性問題自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]中解決.在接下來的第五到第七章中,我們主要對(duì)5度1-傳遞和2-傳遞Cayley圖進(jìn)行了研究.在第五章中,我們通過分類所有無核的5度1-傳遞Cayley圖,證明了周進(jìn)鑫和馮衍全2010年在[On symmetric graphs of valency five, Discrete Math.310 (2010),1725-1732.]上的一個(gè)主要結(jié)論,即：所有非交換單群上的5度1-傳遞Cayley圖都是正規(guī)的.此外,在研究無核的5度弧傳遞Cayley圖的過程中,我們找到了一個(gè)2-弧傳遞的Cayley圖Cay(G,S),滿足Aut(G,S)在S上傳遞但非2-傳遞.從而回答了李才恒2008年在[On automorphism groups of quasiprimitive 2-arc transitive graphs, J Algebr Comb 28 (2008),261-270.]上提出是否存在這樣Cayley圖的問題.在第六章中,我們對(duì)非交換單群上的5度2-傳遞Cayley圖的正規(guī)性進(jìn)行了研究.證明了：除了交錯(cuò)群A39,A59,A119之外的所有非交換單群上的5度2-傳遞Cayley圖都是正規(guī)的.另外,還對(duì)非交換單群上具有可解點(diǎn)穩(wěn)定子群的5度弧傳遞Cayley圖的正規(guī)性進(jìn)行了研究.證明了：除了交錯(cuò)群A39和A79之外的所有非交換單群上具有可解點(diǎn)穩(wěn)定子群的5度弧傳遞Cayley圖都是正規(guī)的.在第七章中,我們分別構(gòu)造了交錯(cuò)群A39,A59,A79和A119上的5度弧傳遞Cayley圖非正規(guī)的例子.在最后一章中,我們對(duì)無平方因子階或者2倍無平方因子階的5度弧傳遞圖進(jìn)行了分類.
[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.
【學(xué)位授予單位】：云南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O157.5
[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.
【學(xué)位授予單位】：云南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O157.5
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