Generalized Quadrangles

This is an outline for a short course on generalized quadrangles. The secret aim is to provide an introduction to many interesting topics from algebraic graph theory. The explicit aim is to provide a counter-example to a proposed algorithm for testing isomorphism of strongly regular graphs.

Actual Topics

  1. Axioms and incidence structures. Examples: $L(K_{m,n}$, edges and 1-factors of $K_6$) Incidence graphs, bipartite graphs with diameter 4 and girth 8. Duality.
  2. Degenerate cases. Thick GQs are semiregular.
  3. Symplectic GQs. Are they self-dual?
  4. Point and line graphs of thick GQs are strongly regular. What does this mean?
  5. SRGS: parameters, eigenvalues, spectral decomposition.
  6. $s\le t^2$, Krein bounds.
  7. known families of GQs, unitary GQs.
  8. ovoids and spreads, covers.
  9. flock GQs.

The standard source for GQs is Payne and Thas "Finite Generalized Quadrangles", but this has a lot that we do not need, and does not cover some material of interest to us. There is a short introduction in Godsil & Royle. I'll add add some more links as we go.