On Study of Multiset Dimension in Fuzzy Zero Divisor Graphs Associated with Commutative Rings

In this paper, we introduce the concept of fuzzy zero divisor graph (FZDG) for a commutative ring \(R\) $R$ denoted by \({\Gamma }_{f}\left(\text{R}\right)\) ${\mathrm{\Gamma }}_f\left(\text{R}\right)$ . We explore the multiset dimension (Mdim), a new variant of the metric dimension (MD), specifically in the context of FZDGs. To illustrate our findings, we analyze the FZDG for the ring \({\mathbb{Z}}_{n}\) ${\mathbb{Z}}_n$ of integers modulo \(n\) $n$ of integers modulo \(n\) $n$ , denoted by \({\Gamma }_{f}\left({\mathbb{Z}}_{n}\right).\) ${\mathrm{\Gamma }}_f\left({\mathbb{Z}}_n\right).$ We compute the multiset dimension for all possible values of \(n\) $n$ for the FZDG \({\Gamma }_{f}\left({\mathbb{Z}}_{n}\right)\) ${\mathrm{\Gamma }}_f\left({\mathbb{Z}}_n\right)$ , providing significant theoretical insights into its structure. Our results not only advance the understanding of FZDGs and their multiset dimensions but also have practical implications across various fields, including cryptography, coding theory, and network analysis. This study lays the groundwork for future research on the application of fuzzy concepts in graph theory and algebraic structures.