The graphs induced by maximal totally isotropic flats of affine-unitary spaces

Let δ=0 or 1, and let AUG(2ν+δ,Fq) be the (2ν+δ)-dimensional affine-unitary space over a finite field Fq. Define a graph Γ whose vertex-set is the set of all maximal totally isotropic flats of AUG(2ν+δ,Fq), and in which F1, F2 are adjacent if and only if dim(F1∪F2)=ν+1, for any F1,F2∈Γ. We show that the distance between any two vertices in Γ is determined by means of dimension of their join and show that Γ is a vertex transitive graph with diameter ν and valency . We also show that any maximal clique in Γ can be changed under the group AU2ν+δ(Fq) into the maximal clique Ω1 with size q(q1/2+1), the maximal clique Ω3 with size qν+δ (δ=0 or 1), or the maximal clique Ω2 with size q3/2+1 (δ=1), and compute the number of maximal cliques containing a fixed vertex in Γ, and the total number of maximal cliques in Γ.