Generalized quadrangle

A generalized quadrangle is by definition a polar space of rank two.

A generalized quadrangle with parameters (s,t) is often denoted by GQ(s,t).

The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation was dubbed "the doily" by Stan Payne in 1973.

There are two interesting graphs that can be obtained from a generalized quadrangle.

[1][2] When looking at the different cases for polar spaces of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles : The generalized quadrangle derived from

Apart from that, only the following parameters have been found possible until now, with q an arbitrary prime power :

GQ(2,2), the Doily
Line graph of generalized quadrangle GQ(2,4)