Some Families Of Arc-transitive Graphs

The main purpose of this thesis is to study two problems associated with arc-transitive graphs in algebraic graph theory. One is to investigate 2-arc tran-sitive graphs which contain an arc-regular action, the other is to investigate the pentavalent arc-transitive graphs under various conditions.In the first chapter 3 and chapter 4, we give a general characterization of the (X,2)-arc transitive, (G, 1)-regular graphs, where G(?)X and X is quasiprimi-tive on the vertex of the graphs. We prove that such graphs are of type PA, AS or TW. For such graphs, we give a family of quasiprimitive 2-arc transitive graphs of product action type and a family of quasiprimitive 2-arc transitive graphs of almost simple type, or some’sporadic’examples. In particular, examples of product action type seem relatively hard to construct, and the existence problem was unsettled until 2006 when some examples were constructed by Cai Heng Li and A. Seress in [Constructions of quasiprimitive two-arc transitive graph-s of product action type, Finite Geometries, Groups and Computation (2006), 115-124] since Praeger posed in 1992.From Chapter 5 to Chapter 7, we investigate pentavalent 1-transitive or 2-transitive Cay ley graphs. In Chapter 5, a characterization of connected core-free pentavalent 1-transitive Cayley graphs is given and we apply the classi-fication result to give another proof for a main result in Jin Xin Zhou and Yan Quan Feng in [On symmetric graphs of valency five, Discrete Math.310 (2010),1725-1732.] which says that all connected pentavalent 1-transitive Cay-ley graphs of finite non-abelian simple groups are normal. Furthermore, in the study of core-free pentavalent arc-transitive Cayley graphs, we find an example of 2-arc transitive Cayley graph Cay(G, S) such that Aut(G, S) is transitive but not 2-transitive on S. It therefore gives a positive answer to a question posed by Cai Heng Li in 2008 in [On automorphism groups of quasiprimitive 2-arc transitive graphs, J Algebr Comb 28 (2008),261-270.].In Chapter 6, we investigate the normality problem of the pentavalent 2-transitive Cayley graphs on finite nonabelian simple groups. We prove that for except the alternating group A39, A59, A119 the pentavalent 2-transitive Cayley graphs on other finite nonabelian simple groups are normal. Further, we also in-vestigate the normality problem of the pentavalent arc-transitive Cayley graphs with soluble stabilizer on finite nonabelian simple groups. We prove that for ex-cept the alternating group A39, A79 the pentavalent arc-transitive Cayley graphs with soluble stabilizer on other finite nonabelian simple groups are normal.In Chapter 7, we construct the examples of nonnormal pentavalent arc-transitive Cayley graphs on the alternating group A39, A59, A79 or A119, respec-tively.In the final Chapter, we classify the pentavalent arc-transitive graphs of square free order or twice a square free order.