Finite S-arc Transitive Graphs

In the thesis we investigate the classification problem of finite s-arc transitive graphs.LetΓbe a graph with vertex set VΓ, edge set EΓand full automorphism group AutΓ. An s-arc inΓis a sequence (α0,α1,…,α8) of vertices inΓsuch that {αi₁,αi}∈EΓandαi-1≠αi+1 for all possible i. A regular graphΓis said to be (G, s)-arc tansitive if G≤AutΓacts transitively on the set of s-arcs; andΓis said to be (G,s)-transitive if it is (G, s)-arc transitive but not (G, s+1)-arc transitive.The study of finite s-arc transitive graphs originated from a beautiful result of W.T. Tutte in 1947:cubic graphs are at most 5-arc transitive. Since then, the study of finite s-arc transitive graphs has received considerable attention in lots of literatures, and gradually become a hot topic of algebraic graph theory. In particular, the classification and characterization of such graphs has been closely watched, which is recognized hard but of great theoretical. In the thesis we focus on research the 2-arc transitive graphs of odd order and cubic s-arc transitive Cayley graphs.A result of C.E. Praeger [57] in 1992 said that every 2-arc transitive graph is a normal cover (see section 2.2) of a quasiprimitive or biquasiprimitive 2-arc transitive graph, and further every quasiprimitive automorphism group is necessary to be one of the four quasiprimitive types. A graph is called a basic graph if it has no nontrivial normal quotient graph. It is well known that the core of the study 2-arc transitive graphs is to classify and characterize the basic one. In case of odd order, C.H. Li [42] proved that every symmetric graph is at most 3-arc transitive and such basic graphs can be constructed from almost simple groups. Then a natural problem arises:to classify and characterize 2-arc transitive graphs of odd order, which is not solved until now. A main work of the thesis is to classify 2-arc transitive basic graphs with odd order. In the course of our work, there are two important tools:one is Thompson-Wielandt Theorem and the other one is the classification of primitive permutation groups of odd degree, which was given by Liebeck and Saxl in 1985.LetΓbe a (G,2)-arc transitive graph of odd order for G is an almost simple group. Note that GαΓ(α), which is the induced permutation group of the stabilizer Gαofα∈VΓonΓ(a) (the neighborhood of a), is a 2-transitive permutation group. Now the 2-transitive permutation groups have been complete classified. At first, by the Thompson and Wielandt'Theorem, we get the general structure of Gα, in particular the insoluble composition factors of Ga. And then, we recur-sively get the accurate structure of G and Ga by making use of the classification theorem of odd degree primitive permutation group and a specific subgroup chain between Gαand G. For the known pairs (G, Gα), we determine the existence of graphs, and further a classification and characterization of 2-arc transitive basic graphs with odd order is given.The other main work of the thesis is to study the cubic symmetric Cavley graphs. It can be proved that every symmetric Cayley graph is either at most 2-arc transitive or a normal cover of a core-free s-arc transitive Cayley graph. C.H. Li has proved that if s≥3, then there are only finite number of core-free s-transitive Cayley graphs. In the thesis, by studying the transitive permutation groups whose degree is not greater than 48, we classify cubic symmetric core-free Cayley graphs and prove that every connected cubic symmetric Cayley graph is normal, or has a normal semiregular subgroup which has exactly two orbital on vertices, or is a normal cover of one of 15 graphs:two 2-transitive graphs, three 3-transitive graphs, four 4-transitive graphs and six 5-transitive graphs. Simultaneously, our argument leads to a well-known result by another proof which is independent of the classical theorem of finite simple groups:all connected cubic symmetric Cayley graphs of finite non-abelian simple groups are normal except two 5-transitive graphs.