Triangle-free distance-regular graphs

Abstract

Let Gamma denote a distance-regular graph with diameter d >= 3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Gamma such that partial derivative(x,y)=partial derivative(z,w)=1, partial derivative(x,z)=3, and partial derivative(x,w)=partial derivative(y,w)=partial derivative(y,z)=2, where partial derivative denotes the path-length distance function. Assume that Gamma has intersection numbers a(1)=0 and a(2)not equal 0. We prove that the following (i) and (ii) are equivalent. (i) Gamma is Q-polynomial and contains no parallelograms of length 3; (ii) Gamma has classical parameters (d,b,alpha,beta) with b <-1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)(2)(b+2)/c (2), (b-2)(b-1)b(b+1)/(2+2b-c (2)) is an integer and that c (2)<= b(b+1). This upper bound for c (2) is optimal, since the Hermitian forms graph Her(2)(d) is a triangle-free distance-regular graph that satisfies c (2)=b(b+1).