The Zero-divisor Semigroups Determined By Some Simple Graphs

The zero-divisor graph is one important subject of the algebraic graph theory. The study on the zero-divisor graphs of semigroups becomes quiet active recently. If a semigroup is given, then we can describe and analyze its graph (fox example, diameter, girth, clique numbers, endomorphism and so on). On the contrary, if a graph is given, and its zero-divisor semigroups exist, then we can characterize the structure of the semigroups, and under some conditions, we can calculate the number of mutually non-isomorphic commutative zero-divisor semigroups. In this paper, we mainly research the zero-divisor semigroups determined by some common graphs on the base of complete graph.In chapterâ… and chapterâ…¡, we mainly introduce some research backgrounds, preliminaries and the conditions under which the zero-divisor graph of semigroup exists.In chapterâ…¢, chapterâ…£and chapterâ…¤, we respectively discuss and characterize the structures of the semigroups determined by the graphs: K*_n+1,K_n·K_m,C₄,K_n∧K_m,K_n*K_m, and under some conditions give corresponding formulasto calculate the number of mutually non-isomorphic commutative zero-divisor semigroups determined by these graphs.