Zero-divisor graphs of Nagata rings and Serre's conjecture rings

Let $R$ be a commutative ring with identity and $X$ an indeterminate over $R$. This project deals with the zero-divisor graphs of the Nagata ring $R(X)$ and the Serre's conjecture ring $R\langle X\rangle$, which are two localizations of the polynomial ring $R[X]$ at the polynomials of unit content and at the monic polynomials, respectively. Our objective is to characterize when the graphs $\Gamma(R(X))$ and $\Gamma(R\langle X\rangle)$ are complete, and investigate their respective diameters and girths. We plan to illustrate the main results with new and original examples.