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.