【作者】 刘慧；

【导师】 冯衍全；

【作者基本信息】 北京交通大学 ， 运筹学与控制论， 2010， 硕士

【摘要】 给定有限群G和它的一个满足S=S-1={s-1|s∈S}和1(?)S的子集S.群G关于S的Cayley图Cay(G,S)定义为具有顶点集G和边集{{g,h}|g,h∈G,gh-1∈S}的图.给定g∈G,定义R(g):h(?)hg,(?)h∈G.映射R:g(?)R(g),(?)g∈G为一个同态映射,其同态像称为群G的右正则表示,记为R(G).易证,R(G)为Aut(Cay(G,S))的正则子群.若R(G)在Aut(Cay(G,S))中正规,则称Cayley图Cay(G,S)为正规的.设n为正整数,Cn[2K1]表示长为n的圈Cn和两点空图2K1的字典式积,即顶点集合为{xi,yi|i∈Zn},边集合为{{xi,xi+1},{yi,yi+1},{xi,yi+1},{yi,xi+1}|i∈Zn}的图.本文主要围绕Cn[2K1]的自同构群展开研究.得到了如下结论：(1)当n=4时,Aut(Cn[2K1])≌S42(?)Z2,当n>4时,Z2n(?)D2n;(2)决定了所有群G及其Cayley子集S使得Cn[2K1]≌Cay(G,S).更多还原

【Abstract】 Let G be a finite group and S a subset of G such that S=S-1= {s-1|s∈S)and 1(?)S.The Cayley graph Cay(G,S)on G with respect to S is defined with vertex set G and edge set{(g,sg)|g∈G,s∈S}.For g∈G, R(g)is defined by:h(?)hg,(?)h∈G.The map R:9(?)R(g),(?)g∈G is an homomorphism,and its homomorphic image is the right regular representation of G,denote by R(G).Obviously, R(G)is a regular subgroup of Aut(Cay(G,S)).A Cayley graph Cay(G,S)is called normal if the right regular representation R(G) of G is a normal subgroup of Aut(Cay(G,s)).Let n be a positive integer.The graph Cn[2K1]is a lexicographic product of Cn and 2K1 with vertex set{xi,yi|i∈Zn}and edge set{{xi,xi+1},{yi,yi+1},{xi, yi+1},{yi,xi+1}|i∈Zn}.The work in this paper is mainly about the automorphism group of Cn[2K1]. We have the following conclusions:(1)Aut(Cn[2K1])is isomorphic to S42(?)Z2 for n=4,or to Z2n(?)D2n for n>4；(2)we have determined all pairs of group G and Cayley subset S of G satisfying Cn[2K1]≌Cay(G,S).更多还原