D-bounded distance-regular graphs

Let Γ = (X, R) denote a distance-regular graph with diameter D ≥ 3 and distance function δ. A (vertex) subgraph Δ ⊆ X is said to be weak-geodetically closed whenever for all x, y ∈ Δ and all z ∈ X, δ(x, z) + δ(z, y) ≤ δ(x, y) + 1 → z ∈ Δ. Γ is said to be D-bounded whenever, for all x, y ∈ X, x and y are contained in a common regular weak-geodetically closed subgraph of diameter δ(x, y). Assume that F is D-bounded. Let P(T) denote the poset the elements of which are the weak-geodetically closed subgraphs of Γ with partial order by reverse inclusion. We obtain new inequalities for the intersection numbers of Γ; equality is obtained in each of these inequalities iff the intervals in P(T) are modular. Moreover, we show this occurs if Γ has classical parameters and D ≥ 4. We obtain the following corollary without assuming Γ to be D-bounded: COROLLARY. Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and D ≥ 4. Suppose that b < -1, and suppose the intersection numbers a₁ ≠ and c₂ > 1. Then β = α 1 + b^D/1 - b.