An extended generalized quadrangle (EGQ) is a connected point-block incidence structure \({\mathcal G}\) with the property that for any point P the points collinear with P (and different from P) and the blocks incident with P form a generalized quadrangle known as the (point) residue at P. In this paper all examples are finite, and all point residues of \({\mathcal G}\) have the same parameters (s,t). So \({\mathcal G}\) is referred to as an EGQ(s,t).
The authors are concerned in general with EGQ(s,t) having a minimal number of points. The extreme case, where the points of \({\mathcal G}\) consist of any one of its points P together with all the points in the residue at P, is called a one-point extension. They were studied extensively by J. A. Thas, Symp. Math. 28, 127-143 (1986; Zbl 0601.51018). So the present authors concentrate on EGQ’s larger than one- point extensions, but which are still relatively small. Many examples are given. Existence and uniqueness results are discussed, and many of them are analogues of (or are equivalent to) similar results for small strongly regular graphs.
Here is a sample theorem: Suppose that an EGQ(s,t), with s even, has the property that, if a point P is not on a block x, then some point of x is not adjacent to P. Then \(v\geq (s+2)(st+t+1).\) If equality holds, then \(t=1\) or \(t=2\). When \(t=1\), the point graph is the complement of a square lattice graph; when \(t=2\), it is the complement of a triangular graph.