Independent sets of some graphs associated to commutative rings

TL;DR

This paper investigates the properties of independent sets and the independence number in zero-divisor graphs associated with commutative rings, focusing on both the standard and ideal-based variants.

Contribution

It provides new insights into the structure of independent sets and calculates the independence number for zero-divisor graphs related to commutative rings.

Findings

Determined the independence number of zero-divisor graphs for certain classes of rings.

Characterized independent sets in ideal-based zero-divisor graphs.

Established relationships between ring properties and graph independence measures.

Abstract

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with cardinality Let $R$ be a commutative ring with nonzero identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by $\Gamma_I(R)$, is the graph which vertices are the set ${x\in R\backslash I | xy\in I \quad for some \quad y\in R\backslash I\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \in I$. In this paper we study the independent sets and the independence number of $\Gamma(R)$ and $\Gamma_I(R)$.