Divisor graph of complement of Gamma(R)

TL;DR

This paper investigates the conditions under which the complement of the zero divisor graph of a finite commutative ring is a divisor graph, providing new characterizations for local rings, products of local rings, and rings with specific associated prime cardinalities.

Contribution

It proves that the complement of the zero divisor graph is a divisor graph for local rings, certain product rings, and rings with exactly two associated primes, answering an open question.

Findings

Complement of zero divisor graph is a divisor graph for local rings.

It is a divisor graph when R is a product of two local rings with one being an integral domain.

If the number of associated primes of R is two, then the complement graph is a divisor graph.

Abstract

Let overline{\Gamma(R)} be the complement of zero divisor graph of a finite commutative ring R. In this article, we have provided the answer of the question (ii) raised by Osba and Alkam in their paper and prove that overline{\Gamma(R)} is a divisor graph if R is a local ring. It is shown that when R is a product of two local rings, then overline{\Gamma(R)} is a divisor graph if one of them is an integral domain. Also, we prove that if cardinality of Ass(R) = 2, then overline{\Gamma(R)} is a divisor graph.