【作者】 凌波；

【导师】 徐尚进；

【作者基本信息】 广西大学 ， 基础数学， 2013， 硕士

【摘要】 在群与图的研究中,我们研究s-传递图基于W. T. Tutte在1947年得到的一个漂亮的结果以及R. Weiss在1981年得到的一个显著结果.W. T. Tutte证明了：对于一个大于等于6的正整数s,不存在三度s-传递图.此外,R.Weiss证明了不存在s=6或者s≥8的s-传递图.自此,分类小度数的s-传递图成为代数图论领域的热门课题.在本论文中,我们考虑某些度数的1-正则Cayley图及5度2-传递Cay ley图且得到以下结果.称图r是1-正则图,如果r的图自同构群Aut(Γ)作用在其弧集上正则.本文首先在第三章中,给出了具有初等交换点稳定子的8度1-正则Cayley图的一个完全分类.证明了：每一个这样的图或者是正规或者双正规或者为某个商图的正规多重覆盖.在本论文的第四章中,给出了奇素数度1-正则Cayley图的一个完全分类.证明了：每一个这样的图或者是正规的或者同构于一个Bi-Cayley图BCay(N,D)或者为5类无核图的正规覆盖.此外,在本章中,还部分的决定了这5类无核图的同构类数.最后一章中,研究了具有点稳定子群Z2×(Z5：Z4)的5度2-传递Cayley图并且证明了：如果Г为正规2-传递Cayley图,则r正规或者双正规.在本章中,还给出了5度连通无核2-传递Cayley图的一个完全分类,其中Cayley子集S由对合组成且点稳定子同构于Z2×(Z5：Z4).更多还原

【Abstract】 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).更多还原