Classical Distance-Regular Graphs of Negative Type

We prove the following theorem. \emsp;Theorem.Let Γ=(X, R)denote a distance-regular graph with classical parameters(d, b, α, β)and d⩾4.Suppose b<−1,and suppose the intersection numbers a1≠0,c2>1.Then precisely one of the following(i)–(iii)holds. (i)Γ is the dual polar graph2A2d−1(−b). (ii)Γ is the Hermitian forms graph Her−b(d). (iii)α=(b−1)/2,β=−(1+bd)/2,and−b is a power of an odd prime.