Classical distance-regular graphs of negative type

Abstract

We prove the following theorem. Theorem. Let Gamma = (X, R) denote a distance-regular graph with classical parameters (d, b, alpha, beta) and d greater than or equal to 4. Suppose b < -1, and suppose the intersection numbers a(1) not equal 0, c(2) > 1. Then precisely one of the following (i) (iii) holds. (i) Gamma is the dual polar graph (2)A(dd-1)(-b). (ii) Gamma is the Hermitian forms graph Her(-b)(d). (iii) alpha = (b - 1)/2, beta = -(1 + b(d))/2, and -b is a power of an odd prime. (C) 1999 Academic Press.