The Pseudo-geometric Graphs for Generalized Quadrangles of Order (3, t)

The values t = 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalised quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalised quadrangle: (v, k, λ, μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 31, 2, 7) and (112, 30, 2, 10). It is well-known that there are two graphs with the first parameter set and that there is just one graph with the last set of parameters. Recently, the second author has shown that there are precisely 28 strongly regular graphs with the second parameter set. Non-existence of a strongly regular graph with the fourth set of parameters was proved by the first author. Here we complete the classification by announcing that there are exactly 167 non-isomorphic strongly regular graphs with parameters (64, 18, 2, 6).