CLASSIFICATION OF COMMUTATIVE ZERO-DIVISOR SEMIGROUP GRAPHS

Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph Gamma(S), whose verticles in Gamma(S) are adjacent if and only of the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether of not ot can be associated to some commutative semigroup, has proved to be a difficult one. In this paper, we extend results by DeMEyer [3], McKenzie and Schneider [4] on this problem by studying the complement of graphs. As an application and an extension of work in [3] we prove that every compact connected 2-manifold admits an Eulerian triangulation that can be associated to a zero divisor semigroup graph.