Zero divisor graphs of semigroups

Abstract

The zero divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. We continue the study of this construction and its extension to a simplicial complex.  2004 Elsevier Inc. All rights reserved. This article continues the study of the zero divisor graph of a commutative semigroup begun (implicitly) in [1,2] and in [4,5], though it is mostly self contained. Throughout S denotes a commutative semigroup with 0 whose operation is written multiplicatively. Associate to S a simple graph G whose vertices are the nonzero zero divisors of S with x = y connected by an edge in case xy = 0. Since the zero divisors of S form an ideal in S, we usually assume S consists of zero divisors. Observe though, that an ideal in a zero divisor semigroup may not consist of zero divisors. For example, If S = {0, x, y | x 2 = x, y 2 = y, xy = 0} then S consists of zero divisors but the ideal {0, x} does not. Recall the semigroup S is nilpotent in case for each x ∈ S there is a positive integer n with x n = 0. Every subsemigroup of a nilpotent semigroup consists of zero divisors. Moreover, in order that G be non empty, we usually assume S always contains at least one nonzero

Unit-Duo Rings and Related Graphs of Zero Divisors

“…The zero-divisor graph of a noncommutative ring (resp. a semigroup) has studied by Redmond and Wu (resp., F. DeMeyer and L. Demeyer) in [12,13,15] (resp., [5]). Zero divisor graph is very useful to find the algebraic structures and properties of rings.…”

The Zero-Divisor Graph Under a Group Action in a Commutative Ring

“…a semigroup) has studied by Redmond and Wu (resp. F. DeMeyer and L. Demeyer) in [9,10,11] (resp [5]). Zero-divisor graph is very useful to find the algebraic structures and properties of rings.…”

A Construction of Commutative Nilpotent Semigroups

“…Since then, much work has been done and this becomes a lively branch in semigroup theory and graph theory, see e.g. [4,5,10,11,12].…”

“…Recall that a simple graph G is called a refinement of a connected simple graph H if V (G) = V (H) and a − b in H implies a − b in G for all distinct vertices of G, where a − b means that a = b and a is adjacent to b. A vertex c is called a center of a graph G if c is adjacent to every vertex of G. In [4,Theorem 3(3)], it was proved that any refinement of a star graph has a corresponding semigroup.…”