Subgraphs Graph in a Distance-regular Graph

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We show G is strongly regular and determine its parameters. Furthermore assume D is at least 4 and set q := bD−1/(bD−2 − bD−1) in the expression of intersection numbers of Γ. We show that G(∆) is Jq(D − s, 2), the Johnson graph or its q-analogue, for each weak-geodetically closed subgraph ∆ of Γ with diameter s at most D− 3 if and only if (bs+1− bs+2)/(bs − bs+1) = q for 0 ≤ s ≤ D − 3, and in this case q is either 1 or a fixed power of a fixed prime.