Weak-Geodetically Closed Subgraphs in Distance-Regular Graphs

We show that if the intersection number c2 > 1 then any weak-geodetically closed subgraph of X is distance-regular. Γ is said to be i-bounded, whenever for all x, y ∈ X at distance δ(x, y) ≤ i, x, y are contained in a common weakgeodetically closed subgraph of Γ of diameter δ(x, y). By a parallelogram of length i, we mean a 4-tuple xyzw of vertices in X such that δ(x, y) = δ(z, w) = 1, δ(x,w) = i, and δ(x, z) = δ(y, z) = δ(y, w) = i− 1. We prove the following two theorems. Theorem 1. Let Γ denote a distance-regular graph with diameter D ≥ 2, and assume the intersection numbers c2 > 1, a1 6= 0. Then for each integer i (1 ≤ i ≤ D), the following (i)-(ii) are equivalent.