On Some1-Regular Or2-Transitive Cayley Graphs

In the study of groups and graphs theory, we are interested in s-transitive graphs because of a beautiful result of W. T. Tutte (1947) and a remarkable result obtained by R. Weiss (1981). W. T. Tutte proved that, for a positive integer s, there exist no s-transitive cubic graphs for s≥6. Moreover, R. Weiss proved that there exist no s-transitive graphs for s=6and s>8. Since then, characterizing s-transitive graphs of small valencies is a active topic in algebraic graph theory. In this paper we consider1-regular Cayley graphs of some valencies and5-valent2-transitive Cayley graphs and then prove the following results.A graph Γ is called1-regular if its full automorphism group Aut(Γ) acts reg-ularly on its arcs. In chapter three of this paper, it is given that a complete clas-sification of8-valent1-regular Cayley graphs with the vertex stabilizer being el-ementary abelian. We prove that every such graphs is either normal or bi-normal or a normal muti-cover of a normal quotient graph.In chapter four of this paper, we give a complete classification or characteri-zation of1-regular Cayley graphs with valency odd prime. It is shown that every such graphs is either normal or is isomorphic to a Bi-Cayley graph BCay(N, D) or is a normal cover of one of5types core-free graphs. Moreover, we also partially decided the numbers of these5types graphs’isomorphic classes.In the last chapter, we investigate2-transitive Cayley graphs of valency five with vertex-stabilizer Z2x (Z5:Z4) and prove that if Γ is normal2-transitive Cayley graphs, then Γ is normal or bi-normal. We also give a complete classi-fication of core-free connected5-valent2-transitive Cayley graphs with Cayley set consists of involutions and stabilizer of full automorphism on vertex1being isomorphic to Z2x (Z5:Z4).