In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle.

The zero-divisor graphs of rings and semirings

The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and quotient sets. We also prove necessary and sufficient conditions for determining when regular graphs and complete graphs with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that satisfy all known necessary conditions but are not zero-divisor graphs.

The Zero-Divisor Graph Associated to a Semigroup

Let S be a (multiplicative) commutative semigroup with 0 Associate to S a (simple) graph G(S) with vertices the nonzero zero-divisors of S, and two distinct vertices x and y are adjacent if and only if xy = 0 In this survey article, we collect some properties of the zero-divisor graph G(S)

The Zero-Divisor Graph of a Commutative Semigroup: A Survey

Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups

By modifying the existing definition of a compressed zero-divisor graph ΓE(K), we define a compressed zero-divisor graph Θ(K) of a finite commutative unital ring K, where the compression is perform...

Erica Purdy

Papers

Classification of commutative zero-divisor semigroup graphs