  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.
