Reconstructing a generalized quadrangle from its distance two association scheme (Q687002)

Given a translation generalized quadrangle \((GQ)\), there is a so-called ``quasi-regular'' point. The second author [Lond. Math. Soc. Lect. Note Ser. 191, 327-339 (1993)] proved that the points at distance two from a quasiregular point form an association scheme with three classes (one of which is given by collinearity). The present authors show, that the \(GQ\) may be reconstructed from this association scheme. More precisely, given a 3-class association scheme with the appropriate parameters and satisfying a suitable assumption on the size of maximal cliques (in the graph defined by one of the classes) the authors show that it arises from a \(GQ\) by Payne's construction.