A Bipartite Graph Linking Units and Zero-Divisors

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