Let ? = ? ( 2 n , q ) be the dual polar graph of type S p ( 2 n , q ) . Underlying this graph is a 2 n -dimensional vector space V over a field F q of odd order? q , together with a symplectic (i.e.?nondegenerate alternating bilinear) form B : V × V ? F q . The vertex set of ? is the set V of all n -dimensional totally isotropic subspaces of? V . If q ? 1 mod?4, we obtain from ? a nontrivial two-graph Δ = Δ ( 2 n , q ) on? V invariant under P S p ( 2 n , q ) . This two-graph corresponds to a double cover ? ? ? ? on which is naturally defined a Q -polynomial ( 2 n + 1 ) -class association scheme on 2 | V ? | vertices.
