Description And Classification Of Some Symmetric Graphs

The wide application of symmetric graph makes it be a hot topic in the study of algebraic graph theory.In this thesis,we mainly investigate the characterization and classification problems of symmetric digraphs,which include determining the upper bound on s for vertex-primitive s-arc-transitive digraphs and the classifica-tion of undirected Cayley graphs on non-abelian simple groups of valency 7.W.T.Tutte[40]proved that symmetric cubic graphs are s-arc-transitive for s≤5 in 1947,and R.Wiss[41]generalized it to any valency:s≤7 in 1981.Since then,a question of the upper bound on s of s-arc-transitive digraphs has received considerable attention.In sharp contrast with the situation for undirected graphs,there are infinite families of s-arc-transitive digraphs with unbounded s other than directed cycles[35].Then what about the situation of these kind of digraphs for vertex primitive?It was shown by Giudici and Xia that the smallest upper bound on s is the same for those admitting an almost simple group.Giudici,Li and Xia[21]proved that s≤2 for G-vertex-primitive and(G,s)-arc-transitive digraphs for almost simple groups G with socle PSLn(q)in 2019.Pan,Wu and Yin[38]proved that s≤2 for all G-vertex-primitive s-arc-transitive digraphs for almost simple groups G with socle an alternating group.In this paper,we mainly investigate the situation of automorphism groups G are exceptional groups of Lie type,and first we studied two minimum groups:Suzuki and Ree groups.We proved that s≤2 for all G-vertex-primitive s-arc-transitive digraphs for almost simple groups G with socle a Suzuki or Ree group,by analyzing whether these groups have nontrivial homogeneous factorization.In 1996,Li[27]gave a classification of cubic symmetric Cayley graphs on nonabelian simple groups.In 2011,Fang et al.[12]gave a classification of sym-metric Cayley graphs on non-abelian simple groups of valency 5.A nature ques-tion:can we give a classification for these kind of graphs of valency 7 or prime?In this paper,we mainly investigate the symmetric Cayley graphs on non-abelian simple groups of valency 7.We analyzed two cases of whether the action of auto-morphism group is quasiprimitive on the vertex set of these kind of graphs:first,the case of acting quasiprimitive is reduced to the case of almost single group;second,the case of non-quasiprimitive is also reduced to the case of almost single group by considering its quotient graph.Then a complete classification of such graphs is given.