The Orbital Graphs Of Finite Permutation Groups And Their Relevant Edge-transitive Graphs

This doctoral thesis is mainly devoted to study orbital graphs of finite permutation groups and their relevant edge-transitive graphs.Characterizing transitive permutation groups whose suborbits have spe-cific property is one of the fundamental topics in the study of permutation groups.The starting point of this thesis is to characterize transitive permu-tation groups admitting a subdegree which is coprime to its degree.Along with this problem,we consider the following exact questions:(1)Characterizing hexavalent edge-transitive Cayley graphs with order coprime to 6.In Chapter 3,we give a characterisation for such graphs,and construct three infinite families of edge-transitive graphs of valency 6 with arbitrary large vertex-stabilizer.This result provides an approach to characterize edge-transitive graphs of valency twice a prime.(2)Constructing and Characterizing edge-transitive multi-cover of cy-cles.In Chapter 4,we construct a family of edge-transitive multi-cover of cycles and give a combinatorial description for this construction.Praeger and Xu in[A characterization of a class of symmetric graphs of twice prime valency.European J.Combin.1989,10(1):91-102]gave a characterization of a family of arc-transitive graphs with valency 2p,whose automorphism groups have an abelian minimal normal subgroup such that the induced normal quotient graph is a cycle.Motivated by this result,we consider edge-transitive Cayley graphs of valency 2p,whose automorphism groups have a non-abelian minimal normal subgroup which is non-regular and the normal quotient graph is a cycle.We give a complete classification for such graphs which are restricted to the condition that all prime divisors of vertices number are greater than p.(3)Characterizing hexavalent edge-primitive graphs.In Chapter 5,we prove that all edge-primitive hexavalent graphs are 2-arc-transitive.Using this result,we show that a hexavalant edge-primitive graphs is either the complete graph K6,6,or its automorphism group is almost simple.Considering the relation between edge-primitive graphs and 2-arc-transitive graphs,we prove that an edge-primitive graph of prime valency is 2-arc-transitive.Also,we provide two examples of edge-primitive graph which are not 2-arc-transitive.Additionally,we give a.complete classification of hexavalent 2-arc-transitive basic graphs of odd order.As a corollary,there exists only 2 edge-primitive hexavalent graphs:K7 and G171.(4)Characterizing primitive permutation groups whose degree and all subdegrees are coprime.In Chapter 6.we give a.characterization for such primitive permutation groups.We prove that the O'Nan-Scott type is HA,AS or PA.We mainly discuss the case for PA type and provide a method to compute subdegrees of PA type primitive permutation group.(5)Are there highly arc-transitive digraphs with non-isomorphic in-local and out-local action?In Chapter 7,we give a positive answer for this question,we provide a construction of a family of highly arc-transitive digraphs,we prove that,for“most of" valencies and ??2,there exist infinite many s-arc-transitive digraphs whose in-local action and out-local action are non-isomorphic.