An Application of a Construction Theory of Strongly Closed Subgraphs in a Distance-regular Graph

Let Γ be a distance-regular graph with (c1, b1) = ⋯ = (cr, br) ≠ (cr+1, br + 1) = ⋯ = (c2r, b2r) wherer ≥ 2 andcr+1>1. We prove thatr = 2 except for the casea1 = ar+1 = 0 andcr+1 = 2 by showing the existence of strongly closed subgraphs.