When Ideal-based Zero-divisor Graphs are Complemented or Uniquely Complemented

TL;DR

This paper classifies when ideal-based zero-divisor graphs of commutative rings are complemented or uniquely complemented, providing insights into their algebraic and graph-theoretic properties.

Contribution

It offers a classification criterion for ideal-based zero-divisor graphs being complemented or uniquely complemented in commutative rings.

Findings

Characterization of when the graph is complemented.

Conditions for the graph to be uniquely complemented.

Connections between ring properties and graph complementarity.

Abstract

Let $R$ be a commutative ring with nonzero identity and $I$ a proper ideal of $R$. The {\it ideal-based zero-divisor graph} of $R$ with respect to the ideal $I$, denoted by $\Gamma_I(R)$, is the graph on vertices $\{x \in R\setminus I \mid xy\in I$ for some $y\in R\setminus I \}$, where distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. In this paper, we classify when an ideal-based zero-divisor graph of a commutative ring is complemented or uniquely complemented.