A Bipartite Graph Linking Units and Zero-Divisors

Abstract

Let $R$ be a commutative ring with identity. We introduce a novel bipartite graph $\mathcal{B}(R)$, the \textit{bipartite zero-divisor--unit graph}, whose vertex set is the disjoint union of the nonzero zero-divisors $Z(R{)}^{\ast }$ and the unit group $U(R)$. A vertex $z\in Z(R{)}^{\ast }$ is adjacent to $u\in U(R)$ if and only if $z+u\in Z(R)$. This construction provides an \textit{additive} counterpart to the well-established \textit{multiplicative} zero-divisor graphs. We investigate fundamental graph-theoretic properties of $\mathcal{B}(R)$, including connectedness, diameter, girth, chromatic number, and planarity. Explicit descriptions are given for rings such as ${\mathbb{Z}}_n$, finite products of fields, and local rings. Our results are sharpest for \textit{finite reduced rings}, where $\mathcal{B}(R)$ yields a graphical characterization of fields and serves as a complete invariant: $\mathcal{B}(R)\cong \mathcal{B}(S)$ implies $R\cong S$ for finite reduced rings $R$ and $S$. The graph also reveals structural distinctions between reduced and non-reduced rings, underscoring its utility in the interplay between ring-theoretic and combinatorial properties.

10 pages