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 (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) If bm+1 ≥ 2, then t ≤ r - 2[s/3]. (2) If cm ≥ 2, then s ≤ r - 2[t/3]. © 2004 Elsevier Ltd. All rights reserved.