A generalization of dual polar graph of orthogonal space

Abstract

Let δ = 0 , 1 or 2, and let AOG ( 2 ν + δ , F q ) be the ( 2 ν + δ ) -dimensional affine-orthogonal space over a finite field F q . Define a graph Γ δ whose vertex-set is the set of all maximal totally isotropic flats of AOG ( 2 ν + δ , F q ) , and in which F 1 , F 2 are adjacent if and only if dim ( F 1 ∪ F 2 ) = ν + 1 , for any F 1 , F 2 ∈ Γ δ . First, we show that the distance between any two vertices in Γ δ is determined by means of dimension of their join and prove that Γ δ is a vertex transitive graph with diameter ν + [ ( 1 + δ ) / 2 ] and valency ( q ν + δ − 1 ) + q 1 + δ [ ν 1 ] q . Next, we show that any maximal clique in Γ δ is isomorphic to the maximal clique Ω 1 δ ( δ ⩾ 1 ) with size q δ + 1 , the maximal clique Ω 2 δ with size 2 q, or the maximal clique Ω 3 δ with size q ν + δ and also compute the total number of maximal cliques in Γ δ . Finally, we study the connectivity of some subgraphs of Γ δ .