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 ∆ ⊆ X is said to be weak-geodetically closed whenever for all x, y ∈ ∆ and all z ∈ X, δ(x, z) + δ(z, y) ≤ δ(x, y) + 1 −→ z ∈ ∆. Γ is said to be D-bounded whenever for all x, y ∈ X, x, y are contained in a common regular weak-geodetically closed subgraph of diameter δ(x, y). Assume Γ is D-bounded. Let P (Γ) denote the poset whose elements 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 if and only if the intervals in P (Γ) 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 b < −1, and suppose the intersection numbers a1 6= 0, c2 > 1. Then