| In this dissertation, we study the zero-divisor graphs of commutative semigroups with 0. The study of zero-divisor graphs was initiated by Istvan Beck in 1988, when he proposed a method for coloring a commutative ring by associating the ring to a simple graph, the vertices of which were defined to be the elements of the ring, with vertices x and y joined by an edge when xy = 0. In 1999 Anderson and Livingston changed this definition, restricting the set of vertices to the non-zero zero divisors of the ring, and from their paper work has proceeded in two directions. Specifically, Redmond investigated zero-divisor graphs of non-commutative rings, while DeMeyer, McKenzie, and Schneider looked at the zero-divisor graphs of commutative semigroups with 0. It is the second of these investigations on which we focus here. Starting with an overview of the essential information from graph theory, we quickly move to an investigation of the structure of semigroups with regard to their zero divisors, concluding that every semigroup can be partitioned into two subsemigroups, one of which is the set of zero divisors. Chapter III looks at the known results linking commutative semigroups and their zero-divisor graphs; in particular, we look at the results of DeMeyer and DeMeyer that determine a set of sufficient conditions for a given graph to be the zero-divisor graph of a commutative semigroup. Chapter IV focuses on extending these results, determining a larger set of graphs which must be the zero-divisor graph of a commutative semigroup. In Chapter V, we use these results to classify the connected graphs on six vertices as to whether or not each is the zero-divisor graph of a commutative semigroup. To accomplish this, we give specific examples of graphs that can be easily classified using the results of Chapter IV; however, we find that there are still some graphs to which the extended results do not apply, and for which we provide a method to classify them. The complete classification of the graphs on six vertices is given in Appendix 1. For graphs that are the zero-divisor graph of a commutative semigroup, we provide the Cayley table of a commutative semigroup; for those that are not, we provide a contradiction that prevents it from being such. In Chapter VI, we begin by noting the fact that every connected graph on three or four vertices is the graph of a commutative semigroup. In fact, most of the graphs are the zero-divisor graph of more than one commutative semigroup, and in the chapter we give methods for determining, up to isomorphism, all of the commutative zero-divisor semigroups for each of the graphs. The complete list of commutative zero-divisor semigroups for each of the graphs is given in Appendix 2. Finally, in Chapter VII, we extend the results of Redmond regarding ideal-based zero-divisor graphs of a commutative ring to the case of commutative semigroups, and close by commenting on a few properties that result from removing the assumption of commutativity from the semigroups. |
