【作者】 贾鲁军；

【导师】 方新贵；

【作者基本信息】 北京大学 ， 基础数学， 2007， 博士

【摘要】 在图的对称性研究中,确定图的自同构群是具有基本重要性的工作.本文主要研究具有有限几乎单群G-弧传递作用图Γ的全自同构群Aut（Γ）.首先,本文提出了T-正规性的概念,这是Cayley图正规性的自然推广,给出了具有单群T-传递作用的图具有T-正规性的充分条件.利用这一充分条件,证明了度数不大于20或是素数度的具有几乎单群G-弧传递作用的图,除了有限个例外,都具有soc（G）-正规性.特别的,在图的度数为3时,排除了所有的例外.利用T-正规性确定了具有几乎单群弧传递作用图的自同构群结构.然后,本文给出了一种构造具有几乎单群弧传递作用图的方法,利用这种方法构造了Aut（Γ）-非拟本原但G-拟本原的G-弧传递图Γ的两个无限族.据作者所知,目前这样的图只有两个无限族,是分别由李才恒[53]和方新贵等人[36]构造的.最后,作为上述结果的应用,本文确定了3度的G-弧传递图Γ的全自同构群Aut（Γ）,即证明了Aut（Γ） = Ree（q）或者Aut（Γ） =Ree（q）×Z₂,其中G = Ree（q） （q≥27）;并且构造出了所有此类图,其中存在图Γ使得Aut（Γ） = Ree（q）×Z₂.更多还原

【Abstract】 A fundamental problem in determining the structure of a graphΓisthe problem of finding its full automorphism group Aut（Γ）. This work ismainly to investigate the full automorphism group Aut（Γ） ofΓ, given itsalmost simple subgroup G.First, a new concept, namely, T-normal graph is introduced, whichis a natural generalization of the concept of normal Cayley graph. Thenwe consider T-vertex-transitive graphs with T a nonabelian simple groupand obtain a sufficient condition under which one can guarantee thatΓisa T-normal graph. Applying the result to G-arc-transitive graphsΓ, weprove that if the valency v（Γ） ofΓis at most 20 or a prime, thenΓisa soc（G）-normal graph for all but finite possibilities of soc（G）. In par-ticular, if v（Γ） = 3, we eliminate all exceptions, that is, ifΓis a cubicG-arc-transitive graph thenΓis soc（G）-normal. By the T-normality, wedetermine the structure of the automorphism group Aut（Γ） ofΓ.Next, a construction is given for an infinite family of G-arc-transitivegraphs. Using this construction we give two family of quasiprimitve arc-transitive graphs which have non-quasiprimitive full automorphism groups.To the author’s best knowledge, the only two infinite families of suchgraphs are constructed by Li [53] and Fang et al. [36], respectively.Finally, as an application of the above results, we completely deter-mine the automorphism group Aut（Γ） of cubic G-arc-transitive graphΓwith G = Ree（q）（q≥27）, and, namely, we prove that Aut（Γ） = Ree（q） or Aut（Γ） = Ree（q）×Z₂. Moreover, we construct all cubic Ree（q）-arc-transitive graphs, amongst which exits there graphsΓsuch that Aut（Γ） =Ree（q）×Z₂.更多还原