Applications of the retracing method for distance-regular graphs

Theorem. Let Γ be a distance-regular graph of diameter d with r=|{i∣(ci,ai,bi)=(c1,a1,b1)}|≥2 and cr+1≥2. Let m, s and t be positive integers with s≤m, m+t≤d and (s,t)≠(1,1). Suppose bm−s+1=⋯=bm=1+bm+1,cm+1=⋯=cm+t=1+cm and am−s+2=⋯=am+t−1=0. Then the following hold.