Generalized quadrangle

In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with n = 4 and near 2n-gons with n = 2. They are also precisely the partial geometries pg(s,t,α) with α = 1. A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certain axioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axioms are the following: There is an s (s ≥ 1) such that on every line there are exactly s + 1 points. There is at most one point on two distinct lines. There is a t (t ≥ 1) such that through every point there are exactly t + 1 lines. There is at most one line through two distinct points. For every point p not on a line L, there is a unique line M and a unique point q, such that p is on M, and q on M and L. (s,t) are the parameters of the generalized quadrangle. The parameters are allowed to be infinite. If either s or t is one, the generalized quadrangle is called trivial. For example, the 3x3 grid with P = {1,2,3,4,5,6,7,8,9} and B = {123, 456, 789, 147, 258, 369} is a trivial GQ with s = 2 and t = 1. 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 has been dubbed "the doily" by Stan Payne in 1973. There are two interesting graphs that can be obtained from a generalized quadrangle. The collinearity graph having as vertices the points of a generalized quadrangle, with the collinear points connected. This graph is a strongly regular graph with parameters ((s+1)(st+1), s(t+1), s-1, t+1) where (s,t) is the order of the GQ. The incidence graph whose vertices are the points and lines of the generalized quadrangle and two vertices are adjacent if one is a point, the other a line and the point lies on the line.