Bicyclic Graphs With Regular Endomorphism Monoids

In the great range of special classes of semigroups, regular semigroups take a central position from the point of view of the richness of their structural "regularity". It seems difficult to obtain a general answer to this question of the characterization of all graphs with regular monoids. As a result, the strategy for answering this question is to find various kinds of conditions for various kinds of graphs. Nowadays many interesting results concerning graphs and their endomorphism monoids have been obtained. In this paper we will give an answer to this question in the range of the bicyclic graphs. This dissertation consists of four chapters. We characterize the endomorphism monoids of bicyclic graphs with two cycles Cn and C’m.In chapter 2,the endomorphism monoid of Gn,m;d(Cn,Cn’)with two disjoint cycles is explored.It is shown that Gn,m;d(Cn,Cn’)is End-regular if and only if n=m=2t+1, t∈N*.as well as |i-j|≠1,i+j≠n+1 and i+j≠n+3 for d(Cn,Cn’)=1 or there is no vertices attached at P2 and i≠j,|i-j|≠2,i+j≠n,i+j≠n+2,i+j≠n+4 for d(Cn,Cn’)=2.In chapter 3,the regularity of Gn,m;Pr is characterized by means of attaching pen-dent vertices at Cn,m;Pr We proved that Gn,m;Pr is End-reegular if and only if it satisfies one of the following conditions:n=m=2t+1,t∈N* and i≠j,i+j≠n+2 for any r≥1 or n=2t+1,t∈N*.,m=4,r=2 and j=3,i≠1,3;j=4,i≠2,n or n=m=4,r=3 and j=4,i≠2,4.