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

Abstract

Let Γ be a distance-regular graph of diameter d with classical parameters (d,b,α,β) with b<−1. Van Dam, Koolen and Tanaka [Distance-regular graphs, Electron. J. Combin. (2016) DS22] surveyed the classification of Γ. 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 Γ. In this paper, we study the above four cases. Our main results are as follows. For the case (1), Γ is the triality graph ▪; for the case (2), Γ 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 Γ are realized by the triality graphs ▪; for the case (3), Γ does not exist; for the case (4), precisely one of the following (i)–(iv) holds: (i) Γ is the dual polar graph ▪, where a1+1 is a prime power; (ii) Γ is the extended ternary Golay code graph; (iii) Γ is the large Witt graph M24; (iv) Γ is the collinearity graph of a maximal regular near hexagon with classical parameters (d,b,α,β)=(3,−a1−1,−c2+a1a1,(a1+1)(c2+a1+1)) with a1≥2, 2≤c2≤(a1)2+a1+1. In particular, if 2≤a1≤107−1, then c2=2 and a1∈{3,4,7,10,17,22,31,52,157}.