A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs

This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as $$, where nonzero elements $$ and $$ are identified as zero divisors if their product results in zero $$ The set of zero divisors in ring $$ is referred to as $$. To analyze various algebraic properties of $$, a graph known as the zero-divisor graph is constructed using $$ This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of $$. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo $$, denoted as $$ the ring of integers modulo $$, denoted as $$, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.