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

Abstract

Let V denote a d -dimensional vector space over F q . 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 q d , q d and q 3 , and that all remaining maximal cliques M satisfy ❘ M ❘ ⩽ q 2 + 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.