Computation of edge metric dimension of zero divisor graph of matrices

This paper investigated the resolving parameters of the zero-divisor graph (ZDG) associated with the non-commutative ring of $3\times 3$ upper triangular matrices over the field ${\mathbb{Z}}_2$. The structural complexity of non-commutative matrix rings, especially the difference between left and right zero-divisors, poses special difficulties for graph-theoretic characterization, although the metric dimensions of ZDGs for commutative rings have been well known. For this particular graph structure $G=ZDG[M_3({\mathbb{Z}}_2)]$, we specifically calculated the metric dimension ( ${\text{dim}}_v(G)$) and the edge metric dimension ( ${\text{edim}}_e(G)$). We determined the minimal resolving sets and proved that ${\text{dim}}_v(G)=[11]$ and ${\text{edim}}_e(G)=[13]$ by combining combinatorial proofs with structural decomposition into equivalence classes. A basic framework for calculating the metric dimensions of generalized $n\times n$ matrix rings over finite fields is provided by these findings.