One of the structures investigated in finite geometry are related to quadratic forms over finite fields (see below for definitions). Knowledge on the geometry of singular points of quadratic forms is very common and covered in many textbooks on finite geometry, but one cannot say the same for the geometry on non-singular points. This short post tries to amend this a little.
(This is no surprise as the geometry with singular points is much nicer than the geometry associated with various types of non-singular points. Also, everything in the following is well-known for decades. It is simply a bit more obscure than other facts about finite quadrics. Lastly, my title is a terrible pun on slang from the mid-20th century (and Pulp Fiction).)
Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36, 10, 4, 2), but there are 32548 non-isomorphic graphs with parameters (36, 15, 6, 6).
Peter Cameron, Random Strongly Regular Graphs?
This a shorter version of this report which I just put on my homepage. But I added more links. I assume that one is familiar with strongly regular graphs (SRGs). One particular SRG, the collinearity graph of , has parameters . A very simple technique, Godsil-McKay (GM) switching, can generate many non-isomorphic graphs with the same parameters. More specifically, there are probably billions such graphs and I generated 13 505 292 of them. This is the number of graphs which you obtain by applying a certain type of GM switching (i.e. using a bipartition of type 4, 59) at most 5 times to . Plots of the number of cliques, cocliques, and the size of the autmorphism group are scattered throughout this post.