Studies On Cayley Graphs And г Graphs Of Semigroups

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.