On a Characterization of Bilinear Forms Graphs

We show that the bilinear forms graphsHq(n, d) of diameterd ≥ 3 are characterized as distance-regular graphs by their parameters provided that eithern ≥ d + 3 andq ≥ 3, orn ≥ d + 4 andq = 2. As a corollary of the method used, we can show the following. If Γ is a distance-regular graph with classical parameters (d, q, α, β) and diameterd ≥ 3, then either Γ is a Johnson graph, a Grassmann graph, a Hamming graph, or a bilinear forms graph, or β is bounded in terms ofd, qand α.