A Graph-Theoretic Approach to Ring Analysis: Dominant Metric Dimensions in Zero-Divisor Graphs

TL;DR

This paper explores the dominant metric dimensions of zero-divisor graphs in finite commutative rings, establishing bounds and analyzing specific ring cases to deepen understanding of their structural properties.

Contribution

It introduces bounds for the dominant metric dimension of zero-divisor graphs and analyzes these dimensions for various specific rings, linking graph properties to algebraic structure.

Findings

Bounds for dominant metric dimension in terms of graph parameters

Analysis of zero-divisor graphs for Gaussian integers modulo m

Insights into rings with identical metric dimensions

Abstract

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 R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x.y=0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. 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. Finally, we provide insights into commutative rings that share identical metric dimensions and dominant metric dimensions. This exploration contributes to a deeper understanding of the structural characteristics of ZD-graphs and their implications for the algebraic properties of commutative rings.