On Certain Bounds for Multiset Dimensions of Zero-Divisor Graphs Associated with Rings

TL;DR

This paper establishes bounds for the multiset dimension of zero-divisor graphs associated with various rings, enhancing understanding of algebraic structures through graph theoretical analysis.

Contribution

It provides new bounds for multiset dimensions in zero-divisor graphs across different rings, including Z_n, Gaussian integers, and quotient polynomial rings.

Findings

Bounds for Mdim in ZD-graphs are derived.

Mdim behavior under algebraic operations is analyzed.

Bounds relate to diameter and maximum degree.

Abstract

This article investigates multiset dimensions in zero divisor graphs (ZD-graphs) associated with rings. Through rigorous analysis, we establish general bounds for the multiset dimension (Mdim) in ZD-graphs, exploring various commutative rings including the ring Z_n of integers modulo n, Gaussian integers and quotient polynomial rings. Additionally, we examine the behavior of Mdim under algebraic operations and discuss bounds in terms of diameter and maximum degree. This study enhances our understanding of algebraic structures and their graphical representations.