Lattices Associated with Subspaces in \(d\)-Bounded Distance-Regular Graphs

Abstract

Let \(\Gamma = (X, R)\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). For \(0 \leq i \leq i+s \leq d-1\), suppose \(\Delta_i\) and \(\Delta_0\) are subspaces with diameter \(i\) and \(i+s\), respectively, and with \(\Delta_i \subseteq \Delta_0\). Let \(\mathcal{L}(i, i+s; d)\) denote the set of all subspaces \(\Delta’\) with diameters \(\geq i\) such that \(d(\Delta_0 \cap \Delta’) = \Delta_1\) and \(d(\Delta_0 + \Delta’) = d(\Delta’) + s\) in \(\Gamma$ including \(\Delta_0\). If we partial order \(\mathcal{L}(i, i+s; d)\) by ordinary inclusion (resp. reverse inclusion), then \(\mathcal{L}(i, i+s; d)\) is a poset, denoted by \(\mathcal{L}_0(i, i+s; d)\) (resp. \(\mathcal{L}_R(i, i+s; d)\)). In the present paper, we show that both \(\mathcal{L}_0(i, i+s; d)\) and \(\mathcal{L}_R(i, i+s; d)\) are atomic lattices, and classify their geometricity.