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

Abstract: 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.