Applications of the retracing method for distance-regular graphs

In the previous paper (J. Combin. Theory Ser. B 79 (2000) 211) we introduced the retracing method for distance-regular graphs and gave some applications. In this paper, we give other applications of this method. In particular, we prove the following result: m. 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.(1) 1≥2, then t≤r−2[s/3]. 2, then s≤r−2[t/3].