有限几何中的默认距离熵与度量维
Default-Distance Entropy and Metric Dimension in Finite Geometries
AI总结：
本文运用信息论，针对源自有限几何的距离正则图与结合方案，证明了其度量维及类维的下界，并在经典对偶极图等三类图中得到具体指数阶下界。
AI中文摘要：
图中的分辨集是一组地标，其距离向量可区分所有顶点。本文运用信息论，对源自有限几何的距离正则图与结合方案的度量维及类维证明下界。核心思想是，对固定地标，随机对象通常位于一个压倒性可能的距离或关系类中。对于经典对偶极图，在秩和类型固定且q通过容许域阶趋于无穷时，我们证明μ(Γ(q,d,e))=Θ_{d,e}(q^e)，其中d≥2且e>0。下界利用对立作为典型距离；上界则对每个固定数量的(d-1)维奇异子空间，取包含它的所有生成元。对于格拉斯曼图、双线性型图及衰减空间方案，我们得到与已知关联结构相同指数阶的下界。
英文摘要：
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.
