Further study of distance-regular graphs with classical parameters with b <-1

Let Gamma be a distance-regular graph of diameter d with classical parameters (d, b, alpha, /3) with b < -1. Van Dam, Koolen and Tanaka [Distance-regular graphs, Electron. J. Combin. (2016) DS22] surveyed the classification of Gamma. From this survey, the following four cases have not been studied: (1) d >= 3, c2 = a1 =1; (2) d >= 3, c2 =1, a2 = a1 > 1; (3) d = 3, c2 =1, a2 > a1 > 1; (4) d = 3, c2 > 1, a1 >= 1. Here ai, bi, ci (0 <= i <= d) are the intersection numbers of Gamma. In this paper, we study the above four cases. Our main results are as follows. For the case (1), Gamma is the triality graph D 3 4,2(2); for the case (2), Gamma is the collinearity graph of a generalized hexagon of order (a1 + 1, (a1 + 1)3). In particular, if a1 + 1 is a prime power, then the classical parameters of Gamma are realized by the triality graphs D 3 4,2(a1 + 1); for the case (3), Gamma does not exist; for the case (4), precisely one of the following (i)-(iv) holds: (i) Gamma is the dual polar graph A 2 5(a1 + 1), where a1 + 1 is a prime power; (ii) Gamma is the extended ternary Golay code graph; (iii) Gamma is the large Witt graph M24; (iv) Gamma is the collinearity graph of a maximal regular near hexagon with classical parameters (c) 2023 Elsevier B.V. All rights reserved.