A Graph-Theoretical Approach to Ring Analysis: An Exploration of Dominant Metric Dimension in Compressed Zero Divisor Graphs and Its Interplay with Ring Structures

The paper systematically classifies rings based on the dominant metric dimensions (Ddim) of their associated CZDG, establishing consequential bounds for the Ddim of these compressed zero-divisor graphs. The authors investigate the interplay between the ring-theoretic properties of a ring ( R ) and associated CZDG. An undirected graph consisting of vertex set ( Z(R_E)\{[0]}\ =\ R_E\{[0],[1]}), where ( R_E=\{[x]:\ x\in R\} ) and ([x]=\{y\in R:\ \text{ann}(x)=\text{ann}(y)\} ) is called a compressed zero-divisor graph, denoted by ( \Gamma_E(R) ). An edge is formed between two vertices ([x]) and ([y]) of ( Z(R_E) ) if and only if ([x][y]=[xy]=[0]), that is, iff ( xy=0 ). For a ring ( R ), graph ( G ) is said to be realizable as ( \Gamma_E(R) ) if ( G ) is isomorphic to ( \Gamma_E(R) ). Moreover, an exploration into the Ddim of realizable graphs for rings is conducted, complemented by illustrative examples reinforcing the presented results. A recent discussion within the paper elucidates the nuanced relationship between Ddim, diameter, and girth within the domain of compressed zero-divisor graphs. This research offers a comprehensive and insightful analysis at the intersection of algebraic structures and graph theory, providing valuable contributions to the current mathematical discourse.