A Bipartite Graph Linking Units and Zero-Divisors
Shahram Mehry, Ali Eisapoor Khasadan
TL;DR
The paper defines the bipartite zero-divisor–unit graph $B(R)$ on the sets $Z(R)^*$ and $U(R)$ with adjacency when $z+u$ remains a zero-divisor, establishing a new additive perspective on zero-divisor graphs. It provides a comprehensive graph-theoretic study, including tight results for finite reduced rings (notably a complete bipartite characterization, connectedness with diameter at most $4$, and $B(R)$ as a complete invariant in that class), along with analyses of planarity and automorphism groups. A key finding is that, for finite reduced rings, $B(R)\cong B(S)$ implies $R\cong S$, linking graph structure directly to ring structure; the work also clarifies distinctions between reduced and non-reduced rings via the graph's edges and cycles. The framework opens avenues for spectral studies, noncommutative extensions, and deeper exploration of ring equivalence through graphical invariants, highlighting the practical impact of additive, combinatorial perspectives in ring theory.
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 $Z(R)^*$ and the unit group $U(R)$. A vertex $z \in Z(R)^*$ 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.