D-bounded Distance-regular Graphs

Let Γ=(X, R) denote a distance-regular graph with diameterD≥3 and distance functionδ. A (vertex) subgraph Δ⊆Xis said to beweak-geodetically closedwhenever for allx, y∈Δ and allz∈X,δ(x, z)+δ(z, y)≤δ(x, y)+1→z∈Δ.Γis said to beD-boundedwhenever, for allx, y ∈X, xandyare contained in a common regular weak-geodetically closed subgraph of diameterδ(x, y).Assume that Γ isD-bounded. LetP(Γ)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 inP(Γ)are modular. Moreover, we show this occurs if Γ has classical parameters andD≥4.We obtain the following corollary without assuming Γ to beD-bounded: ARY.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≠0 and c2>1. Thenβ=α1+bD1−b