The Unit-Zero Divisor Graph of a Commutative Ring

TL;DR

This paper introduces the unit-zero divisor graph of a commutative ring, a new graph-theoretic framework that captures the interplay between additive and multiplicative properties of the ring.

Contribution

It defines a novel graph structure for rings based on dual adjacency conditions and explores its structural properties and algebraic influences.

Findings

The graph's regularity and bipartiteness depend on ring properties.

Planarity and Hamiltonicity are characterized in relation to ring structure.

The graph reflects the interaction between units, zero divisors, and ideals.

Abstract

This paper introduces a new approach to associating a graph with a commutative ring. Let $R$ be a commutative ring with identity. The unit-zero divisor graph of a commutative ring $R$, denoted by $G_{UZ}(R)$, offers a novel framework for exploring the interaction between ring and graph structures. The vertex set of $G_{UZ}(R)$ consists of all elements of the ring $R$. Two distinct vertices $x$ and $y$ in $G_{UZ}(R)$ are adjacent if and only if $x + y$ is a unit and $xy$ is a zero divisor in $R$. This dual adjacency condition gives rise to a graph that reflects both the additive and multiplicative behavior of the ring. This study investigates key structural properties of $G_{UZ}(R)$, including regularity, bipartiteness, planarity, and Hamiltonicity. In addition, it examines how these graph features are influenced by the algebraic structure of the ring, particularly the group of units, the set of zero divisors, ideals, and the Jacobson radical.