Computation of edge metric dimension of zero divisor graph of matrices

Sahil Sharma; Omaima Al Shanqiti; Vijay Kumar Bhat; Sahil Sharma; Omaima Al Shanqiti; Vijay Kumar Bhat

doi:10.3934/math.2026526

AIMS Mathematics

2026, Volume 11, Issue 5: 12780-12794. doi: 10.3934/math.2026526

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Computation of edge metric dimension of zero divisor graph of matrices

1.
School of Mathematics, Shri Mata Vaishno Devi University, Jammu and Kashmir, Katra 182320, India
2.
Department of Mathematics, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 10 February 2026 Revised: 07 April 2026 Accepted: 28 April 2026 Published: 08 May 2026
MSC : 05C12, 05C76, 05C90

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.
- metric dimension,
- ZDG,
- edge metric dimension,
- upper triangular matrix,
- resolving set
Citation: Sahil Sharma, Omaima Al Shanqiti, Vijay Kumar Bhat. Computation of edge metric dimension of zero divisor graph of matrices[J]. AIMS Mathematics, 2026, 11(5): 12780-12794. doi: 10.3934/math.2026526

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Abstract

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.

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