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

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 $Z(R_E)\mathrm{\backslash }\{[0]\}=R_E\mathrm{\backslash }\{[0],[1]\}$, 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 ${\mathrm{\Gamma }}_E(R)$. 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 ${\mathrm{\Gamma }}_E(R)$ if $G$ is isomorphic to ${\mathrm{\Gamma }}_E(R)$. 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.