Year: 2026
Type: preprint
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 $q\to\infty$ through the admissible field orders, we prove $μ(Γ(q,d,e))=Θ_{d,e}(q^e)$ for $d\geq 2$ 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.
Source: arXiv (Cornell University)
Author: Vazquez, Maximiliano
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204851193