TY - JOUR

T1 - Classification of commutative zero-divisor semigroup graphs

AU - Lisa, Demeyer

AU - Jiang, Yunjiang

AU - Loszewski, Cleland

AU - Purdy, Erica

PY - 2010

Y1 - 2010

N2 - Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph r(S), whose vertices are labeled with the nonzero elements in S. Two vertices in T(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it 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.

AB - Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph r(S), whose vertices are labeled with the nonzero elements in S. Two vertices in T(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it 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.

UR - http://www.scopus.com/inward/record.url?scp=79551597891&partnerID=8YFLogxK

U2 - 10.1216/RMJ-2010-40-5-1481

DO - 10.1216/RMJ-2010-40-5-1481

M3 - Article

AN - SCOPUS:79551597891

VL - 40

SP - 1481

EP - 1503

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 5

ER -