Graph Theoretic Properties of the Zero-Divisor Graph of a Ring

Let R be a commutative ring with 1 ≠ 0, and let Z(R) denote the set of zero-divisors of R. One can associate with R a graph Γ(R) whose vertices are the nonzero zero-divisors of R. Two distinct vertices x and y are joined by an edge if and only if xy = 0 in R. Γ® is often called the zero-divisor graph of R. We determine which finite commutative rings yield a planar zero-divisor graph. Next, we investigate the structure of Γ(R) when Γ(R) is an infinite planar graph. Next, it is possible to extend the definition of the zero-divisor graph to a commutative semigroup. We investigate the problem of extending the definition of the zero-divisor graph to a noncommutative semigroup, and attempt to generalize results from the commutative ring setting. Finally, we investigate the structure of Γ(k1 × ∙ ∙ ∙ × kn) where each ki is a finite field. The appendices give planar embeddings of many families of zero-divisor graphs.