On a new extension of the zero-divisor graph

TL;DR

This paper introduces a new graph structure based on nonzero zero-divisors of a commutative ring, exploring its properties and relationships with existing zero-divisor and total graphs.

Contribution

It proposes a novel graph extension involving zero-divisors, analyzing its properties and connections to established algebraic graphs.

Findings

The new graph generalizes the zero-divisor graph.

It reveals relationships between zero-divisor and total graphs.

Examples illustrate the graph's properties and differences.

Abstract

In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+y\in Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.