Classfication Of Some Symmetric Graphs With Transitive Subgroups

The study of symmetric graphs is always a hot topic in the study of algebraic graph theory.In this paper,we study the characterization and classification of sym-metric graphs with transitive subgroups,mainly the classification of vertex primitive s-transitive Cayley graphs and prime-valent symmetric graphs admitting transitive automorphism alternating group.The study of finite s-transitive graphs can be traced back to a conclusion proved by Tutte in the literature[1]in 1949:every finite symmetric graph of valency 3 is at most 5-arc transitive.Weiss extended this result to the general case in the literature[2]in 1981.He proved that every symmetric graph is at most 7-arc transitive.In 2005,C.H.Li proved that for s ∈{2,3,4,5,7} and k≥3 there are only finite number of core-free s-transitive Cayley graphs of valency k,and that,with the exceptions s=2 and(s,k)=(3,7),every s-transitive Cayley graph is a normal cover of a core-free one.In the same year,he prove that "almost all" of the vertex primitive Cayley graphs are all normal Cayley graphs.So a natural question has aroused widespread concern:the non-normal vertex primitive Cayley graphs be completely classified?This part of this paper is devoted to the characterization of non-normal vertex prim-itive Cayley graphs.Let Γ be a vertex primitive s-arc transitive Cayley graph and X≤Aut(Γ).First,G is reduced to almost simple group,then the structure infor-mation of automorphism group of X is obtained by global analysis method,and the structure information of vertex stabilizer of X is obtained by local analysis method,Finally,by using the structure of maximal subgroups and transitive permutation rep-resentation,the existence of graphs is determined,and the complete classification of such graphs is given.It is still a hot topic to study the classification of symmetric graphs under some limited conditions,for example:M.Y.Xu et al gave the classification of symmetric graphs of order 3p in the literature[4],where p is prime.S.J.Xu et al and C.H.Li et al gave the classification of cubic s-arc transitive Cayley graphs of finite simple groups in the literature[5]and[6].Y.Q.Feng et al and X.G.Fang et al gave the classification of symmetric Cayley graphs of finite simple groups with valency 5 in the literature[7]and[8].Y.Q.Feng et al gave the classification of prime-valent arc-transitive Cayley graphs of finite simple groups with solvable vertex-stabilizer in the literature[9].The motivation of this paper comes from the B.Z.Xia’s result of literature[10],which completely classifies the quasiprimitive groups with a tran-sitive alternating automorphism group.Therefore,the classification of prime-valent symmetric graphs with a transitive alternating automorphism group has become a hot topic.The second part of this paper is to solve the above problem,that is,com-plete classification of prime-valent symmetric graphs with a transitive alternating automorphism group.Let Γ be a prime-valent G-symmetric graph with transitive alternating automorphism group and G≤Aut(Γ).In this paper,we first reduce G to almost simple group,and then give a complete classification of such graphs by using the knowledge of factorization,structure of prime-valent vertex-stabilizer subgroups and transitive permutation representation.