【作者】 杨东；

【导师】 罗彦锋；

【作者基本信息】 兰州大学 ， 基础数学， 2011， 博士

【摘要】 对半群的Cayley图的研究是近年来一个十分活跃的研究领域,本文定义了半群的Cayley图的一种推广图Г图,研究了半群的Cayley图和Г图的结构和性质.设S是一个半群,T1,T2是S的两个子集,且T1与T2中至少有一个是非空集合.称一个有向图为S的Г图,记为Г,如果V(г)=S,E(г)={(u,υ)∈S×S|u≠υ,存在t1∈T11,t2∈T21,使得υ=t1ut2).当T1=T2=S时,S的Г图就是S的除图；当T2=φ时,S的r图就是S的关于联络集T1的Cayley图Cay(S,T1).首先,我们通过半群的Г图研究了半群的一个组合性质：D-饱和性,其中D是一个有限图.给出了半群是ГD-饱和的的充分必要条件,不但推广了A.V.Kelarev和S.J.Quinn关于半群是Cayley图D-饱和的的工作,而且解决了半群什么时候是除图D-饱和的的问题.其次我们刻画了完全单的周期半群的Cayley图的点可迁性,得到了完全单的周期半群的Cayley图Cay(G,S)是ColAuts(G)点可迁的,Ends(G)-点可迁的以及ColEnds(G)-点可迁的的充分必要条件。描述了单演半群、矩形带、正规带、左群及群等一些特殊的半群类的Г图的结构和性质,也研究了一般半群的Г图,得到了若干性质,包括给出了Г图是ColEnd(г)点可迁的必要条件.最后我们研究了偶数阶群的拟交换Cayley图的匹配可扩性,给出了偶数阶群的拟交换Cayley图是2-可扩的完全刻画.更多还原

【Abstract】 The research on Cayley graphs of semigroups is an active scientific field in recent years. In this thesis we define aгgraph: generalization of Cayley graphs and then do some researches on structure and properties of Cayley graphs andгgraphs of semigroups.Let S be a semigroup and T1, T2 subsets of S. Suppose either T1 no empty or T2 no empty. A directed graph is defined asгgraph, denoted byг, of S, if V(г)=S,E(г)={(u,v)∈S×S|u≠v,v=t1ut2 for some t1∈T11,t2∈T21}. In particular, in case of T1=T2=S, theгgraph of S is exactly the Div(S); in case of T2=(?), theгgraph of S is exactly the Cay(S, T1) of S relative to T1.Firstly, we study the D-saturation about semigroups by theгgraphs of semigroups, where D is a finite graph, and also give some necessary and sufficient conditions for semigroups beingгD-saturated. The results not only generalize the results on D-saturation of Cayley graphs of semigroups obtained by A. V. Kelarev and S.J. Quinn, but also give the conditions of semigroups being Div(S) D-saturated.Secondly we characterize the vertex-transitive property of Cayley graphs of completely simple periodic semigroups, obtain necessary and sufficient con-ditions for Cayley graphs of completely simple periodic semigroups being ColAu ts(G)-vertex transitive, Ends (G)-vertex transitive and ColEnds(G)-vertex tran-sitive, respectively, give not only structure and properties ofгgraphs of such special graphs as monogenic semigroups, rectangular bands, normal bands, left groups and groups,but also some properties of general semigroups including the necessary conditions forгgraphs being ColEnd(г)-vertex transitive.Finally, we study match extendability of quasi-abelian Cayley graphs of groups with even order, give a complete characterization for 2-extendability of quasi-abelian Cayley graphs of the groups with even order.更多还原