Applications of the retracing method for distance-regular graphs

Abstract

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: Theorem. Let Γ be a distance-regular graph of diameter d with r=|{ i∣( c i , a i , b i )=( c 1, a 1, b 1)}|≥2 and c r+1 ≥2. Let m, s and t be positive integers with s≤ m, m+ t≤ d and ( s, t)≠(1,1). Suppose b m− s+1 =⋯= b m =1+ b m+1 , c m+1 =⋯= c m+ t =1+ c m and a m− s+2 =⋯= a m+ t−1 =0. Then the following hold. (1) If b m+1 ≥2, then t≤ r−2[ s/3]. (2) If c m ≥2, then s≤ r−2[ t/3].