Classification of the Maximal Cliques of Size ⩾q + 4 m the Quadratic Forms Graph in Odd Characteristic

Let V denote a d-dimensional vector space over Fq. Associated to V is a distance-regular graph Quad(d, q), the vertex set of which consists of all quadratic forms on V and the edges (x, y) of which are defined by the property rk(x - y) = 1 or 2. In [8] it is shown that there are three distinguished classes of maximal cliques of respective sizes qd, qd and q3, and that all remaining maximal cliques M satisfy ❘M❘ ⩽ q2 + q + 2. In the present work, we obtain a strong refinement of this result when q⩾ 5 is odd, providing a complete classification of all maximal cliques of size at least q + 4