On the nonexistence of pseudo-generalized quadrangles

In this paper, we consider the question of when a strongly regular graph with parameters ((s+1)(st+1),s(t+1),s−1,t+1) can exist. A strongly regular graph with such parameters is called a pseudo-generalized quadrangle. A pseudo-generalized quadrangle can be derived from a generalized quadrangle, but there are other examples which do not arise in this manner. If the graph is derived from a generalized quadrangle then t≤s² and s≤t², while for pseudo-generalized quadrangles we still have the former bound but not the latter. Previously, Neumaier has proved a bound for s which is cubic in t, but we improve this to one which is quadratic. The proof involves a careful analysis of cliques and cocliques in the graph. This improved bound eliminates many potential parameter sets which were otherwise feasible.