Default-Distance Entropy and Metric Dimension in Finite Geometries
Abstract
A resolving set in a graph is a set of landmarks whose distance vectors distinguish all vertices. We use information theory to prove lower bounds for metric dimension and class dimension in distance-regular graphs and association schemes arising from finite geometry. The core idea is that, for a fixed landmark, a random object usually lies in one overwhelmingly likely distance or relation class. For classical dual polar graphs, with rank and type fixed and q→∞ through the admissible field orders, we prove μ(Γ(q,d,e))=Θd,e(qe) for d≥ 2 and e>0. The lower bound uses opposition as the typical distance. For the upper bound, we take, for each of a constant number of (d−1)-dimensional singular subspaces, all generators containing it. For Grassmann graphs, bilinear forms graphs, and attenuated-space schemes, we obtain lower bounds of the same exponential order as the known incidence constructions.
Related
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BibTeX
@misc{vazquez2026defaultdistance,
title = {{Default-Distance Entropy and Metric Dimension in Finite Geometries}},
author = {Maximiliano Vazquez},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27788},
eprint = {2608.27788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27788},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Default-Distance Entropy and Metric Dimension in Finite Geometries
AU - Maximiliano Vazquez
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27788
UR - https://arxiv.org/abs/2608.27788
AB - A resolving set in a graph is a set of landmarks whose distance vectors distinguish all vertices. We use information theory to prove lower bounds for metric dimension and class dimension in distance-regular graphs and association schemes arising from finite geometry. The core idea is that, for a fixed landmark, a random object usually lies in one overwhelmingly likely distance or relation class. For classical dual polar graphs, with rank and type fixed and q→∞ through the admissible field orders, we prove μ(Γ(q,d,e))=Θ_d,e(q^e) for d≥ 2 and e>0. The lower bound uses opposition as the typical distance. For the upper bound, we take, for each of a constant number of (d−1)-dimensional singular subspaces, all generators containing it. For Grassmann graphs, bilinear forms graphs, and attenuated-space schemes, we obtain lower bounds of the same exponential order as the known incidence constructions.
ER -