Suborbits of (m,k)-isotropic subspaces under finite singular classical groups

Let be one of the (2ν+δ+l)-dimensional singular classical spaces and let G2ν+δ+l,2ν+δ be the corresponding singular classical group of degree 2ν+δ+l. All the (m,k)-isotropic subspaces form an orbit under G2ν+δ+l,2ν+δ, denoted by M(m,k;2ν+δ+l,2ν+δ). Let Λ be the set of all the orbitals of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)). Then (M(m,k;2ν+δ+l,2ν+δ),Λ) is a symmetric association scheme. First, we determine all the orbitals and the rank of (G2ν+δ+l,2ν+δ,M(m,k;2ν+δ+l,2ν+δ)), calculate the length of each suborbit. Next, we compute all the intersection numbers of the symmetric association scheme (M(ν+k,k;2ν+δ+l,2ν+δ),Λ), where k=1 or k=l−1. Finally, we construct a family of symmetric graphs with diameter 2 based on M(2,0;4+δ+l,4+δ).