Graph-theoretic characterization of rings: Outer multiset dimension of zero-divisor graphs

A finite unital commutative ring (UCR) is denoted by $A$. The elements $\zeta \ne 0,\eta \ne 0$ in $A$ are zero divisors if their product satisfies $\zeta \cdot \eta =0$. The set of zero divisor graph in $A$ is denoted by $D\left(A\right)$. A zero divisor graph is constructed using $D\left(A\right)$ in order to analyze various algebraic properties. In this article, we characterize the rings based on Outer multiset dimension (OMdim) of their associated zero divisor graphs. For this purpose, we study the zero divisor graphs of rings, including the ring of Gaussian integers modulo $m$, ${ℨ}_m\left[i\right]$, the ring of integers modulo $n$, ${ℨ}_n$, and certain quotient polynomial rings. Also particularly, we study the OMdim of zero divisor graphs of ring $Z_n$ for all values of $n$.