Two Classes Of Edge-transitive Graphs With Almost Simple Groups

This thesis mainly studies the automorphism groups of vertex-quasiprimitive edge-transitive graphs and the classification problem of edge-primitive graphs.The symmetry of graphs(such as edge-transitivity,arc-transitivity,etc.)and the automorphism groups of graphs are important objects in algebraic graph theory,and the theory of groups always plays an irreplaceable role in the research.Quite a lot of problems on symmetric graphs have been reduced to the class of graphs admitting quasiprimitive groups or even almost simple groups,especially for those graphs with highly transitive properties.This is the main motivation for the research in this thesis.The first problem we considered in this thesis is about the characterization of vertex-quasiprimitive edge-transitive graphs.Let ?=(V,E)be a connected edge-transitive graph of twice prime valency,where G is a subgroup of the automorphism group of ?.We first analyze the actions of the normal subgroups of G on the graph ?,then investigate the symmetric properties of the normal quotients of ? and the struc-tural information bearing from the original graph ?.Based on these,when G is a quasiprimitive permutation group on the vertex set V,we proved that one of the follow-ing holds:(1)? is a(G,2)-arc-transitive graph,(2)G is an almost simple group,(3)G is an affine primitive group.This result provides us a theoretical support to classify vertex-quasiprimitive edge-transitive graphs of odd order and twice prime valency.For the case where G is a primitive group(on the vertex set V)of odd degree,we proved that G and the automorphism group of ? have the same socle unless ? is a complete graph.If further G is an almost simple primitive group of odd degree,then the socle of G acts transitively on the edge set E unless ? is one of two exceptional graphs with valency 4.The second problem we considered in this thesis is about the classification of edge-primitive graphs.The class of edge-primitive graphs consists of special edge-transitive graphs,which have very strict restrictions on the automorphism groups and local structures.Actually,there are only four O'Nan-Scott types can occur for the automorphism group of an edge-primitive graph acting on the edge set,and the action of the automorphism group on the vertex set is well described too.Especially,under the condition of 2-arc-transitivity,the automorphism group of an edge-primitive graph is an almost simple group unless the graph is either a cycle of prime length or a complete bipartite graph.This reminds us the classification for these 2-arc-transitive and edge-primitive graphs.In this thesis we solved the case when the edge stabilizer is soluble.Let ?={V,E}be a G-edge-primitive and(G,2)-arc-transitive graph of valency d,where G?Aut?,d?3,we proved that,up to isomorphism of graphs,the class of graph G consists of 31 single graphs and 12 infinite families.