A generalization of dual polar graph of orthogonal space

Let δ=0,1 or 2, and let AOG(2ν+δ,Fq) be the (2ν+δ)-dimensional affine-orthogonal space over a finite field Fq. Define a graph Γδ whose vertex-set is the set of all maximal totally isotropic flats of AOG(2ν+δ,Fq), and in which F1, F2 are adjacent if and only if dim(F1∪F2)=ν+1, for any F1,F2∈Γδ. First, we show that the distance between any two vertices in Γδ is determined by means of dimension of their join and prove that Γδ is a vertex transitive graph with diameter ν+[(1+δ)/2] and valency . Next, we show that any maximal clique in Γδ is isomorphic to the maximal clique (δ⩾1) with size qδ+1, the maximal clique with size 2q, or the maximal clique with size qν+δ and also compute the total number of maximal cliques in Γδ. Finally, we study the connectivity of some subgraphs of Γδ.