The spectrum of the second subconstituent of the bilinear forms graph \(H_q(d,e)\) (Q1266387)

Let \(Y\) be a \((d+e)\)-dimensional vector space over the field \(F_q\) with a fixed \(d\)-dimensional subspace \(U\). The vertices of \(H_q(d,e)\) are the \(e\)-dimensional subspaces disjoint from \(U\). Two vertices are adjacent iff they have an intersection of dimension \(e-1\). In this paper the spectrum of the second subconstituent of \(H_q(d,e)\) is obtained. There are \(16\) eigenvalues (when \(d=e\) some eigenvalues collapse), but in the cases \(q=2\), \(d\leq 3\), \(e\leq 3\) some eigenvalues have multiplicity zero.