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arXiv · 2405.04934

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

Nasir Ali·Hafiz Muhammad Afzal Siddiqui·Muhammad Imran Qureshi
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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 ( Γ_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 ( Γ_E(R) ) if ( G ) is isomorphic to ( Γ_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.

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Nasir Ali, Hafiz Muhammad Afzal Siddiqui, Muhammad Imran Qureshi. 2024-05-08. A Graph-Theoretical Approach to Ring Analysis: An Exploration of Dominant Metric Dimension in Compressed Zero Divisor Graphs and Its Interplay with Ring Structures. https://arxiv.org/abs/2405.04934

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