The Unit-Zero Divisor Graph of a Commutative Ring

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.