On a Characterization of Bilinear Forms Graphs

Abstract

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.