Symplectic graphs

Consider a vector space V of dimension d over the finite field with q elements provided with a nondegenerate symplectic form f. (Then d will be even.) Take the points in the projective space PV as vertices of a graph, and join the points u and v when f(u,v) = 0. The resulting graph is denoted by Sp(d,q).

For d=2 the resulting graph is a (q+1)-coclique. For d>2 this graph is strongly regular, with parameters v = (qd − 1)/(q − 1), k = q(qd−2 − 1)/(q − 1), λ = (qd−3 − 1)/(q − 1) + q − 2, μ = (qd−2 − 1)/(q − 1).

A projective line is totally isotropic if it induces a clique in this graph, and conversely every edge spans a totally isotropic line, so that the graph is the collinearity graph of the polar space formed by the projective points and the totally isotropic lines. In particular, the graph Sp(4,q) is the collinearity graph of a generalized quadrangle with parameters GQ(q,q). See also GQ(2,2).