📂Graph Theory

Erdős–Rényi Graph

Definition

Let’s assume a commutative ring $R$ is given. The set of zero divisors within $R$ is denoted as $Z(R)$. The graph $\Gamma (R)$ defined below is referred to as the Zero Divisor Graph for $R$. $$ V \left( \Gamma (R) \right) = Z(R) \\ E( \Gamma (R)) = \left\{ ab : ab=0 \right\} $$

Description

As is known, the product of zero divisors does not necessarily yield $0$. For example, although $ 2, 4 \in Z \left( \mathbb{Z}_{10} \right)$ is a zero divisor of $\mathbb{Z}_{10}$, its product results in $8 \ne 0$. Hence, the zero divisor graph associated with a given commutative ring $R$ takes on non-trivial forms, making its properties or classification a subject of interest. For instance, if we consider $\mathbb{Z}_{12}$, its zero divisor graph $\Gamma \left( \mathbb{Z}_{12} \right)$ is depicted as follows:

History

The history of zero divisor graphs is relatively short. Initially defined by Beck in 1988, research on zero divisor graphs was further advanced by David F. Anderson and Philip Livingston. Anderson, being Livingston’s mentor, contributed greatly to the field with the publication of the Anderson-Livingston Theorem¹, a significant achievement in the study of zero divisor graphs.

Given that the zero divisor graph of an integer ring is the most intuitive example, research has also expanded to include other types of rings. In 2008, Emad Abu Osba of the University of Jordan² classified the zero divisor graph $\Gamma \left( \mathbb{Z}[i] \right)$ of a Gaussian ring, and in 2014, Osama Alkam³ classified the zero divisor graph $\Gamma ( \mathbb{Z} [ \omega] )$ of an Eisenstein ring.