A unified approach to a characterization of Grassmann graphs and bilinear forms graphs (Q1332358)

The Grassmann graph \(J_ q(n,d)\) is defined over the set of all \(d\)- dimensional subspaces of \(\mathbb{F}^ n_ q\). Two vertices \(x\) and \(y\) are adjacent when \(\dim(x\cap y)= d- 1\). The bilinear forms graph \(H_ q(n,d)\) is defined on the set of all bilinear forms on \(\mathbb{F}^ d_ q\times \mathbb{F}^ n_ q\) (with \(d\leq n\)). Two forms \(e\) and \(f\) are adjacent if the rank of \(e-f\) is equal to 1. The purpose of this paper is to give a characterization of these graphs among a family of distance- regular graphs defined by some classical parameters and submitted to some extra geometric conditions.