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

2024/05/10 by Nasir Ali, Ali, Nasir, Hafiz Muhammad Afzal Siddiqui +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2405.06187

openalex publication_date 2024/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ R\ and [x]=\y∈ R : ann(x)=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). 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.

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