Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs

TL;DR

This paper investigates the multiset dimension of compressed zero-divisor graphs linked to rings, classifies rings based on this dimension, and explores bounds and properties of these graphs.

Contribution

It introduces the concept of multiset dimension for compressed zero-divisor graphs and classifies rings accordingly, providing bounds and analyzing graph properties.

Findings

Classification of rings based on multiset dimension

Bounds established for the multiset dimension of CZDG

Analysis of girth and diameter relationships in CZDG

Abstract

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDG) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring $R$ and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set $ Z(R_E)\backslash\{[0]\} = R_E\backslash\{[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 $\Gamma_E (R)$. We classify the rings based on Mdim of their associated CZDG and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Lately, we have discussed the interconnection between Mdim, girth, and diameter of CZDG.