Several anzahl formulas of classical spaces and their applications

Let Fq2ν+δ be one of the (2ν+δ)-dimensional classical spaces and P be a fixed subspace of type ϑ of Fq2ν+δ. Let M(i;ϑ;m;2ν+δ) be the set of all m-dimensional totally isotropic subspaces Q of Fq2ν+δ satisfying dim(P∩Q)=i. In this paper, we compute the size of M(i;ϑ;m;2ν+δ) when P is non-isotropic or totally isotropic. As applications, we give lower bounds of ranks of incidence matrices of totally isotropic subspaces of Fq2ν+δ over the real number field R, and compute eigenvalues of some graphs.