strongly regular graphs

Collecting Strongly Regular Graphs

Today I write about a recent hobby of mine: Collecting strongly regular graphs. It started three years ago. You can find my collection on my homepage. I collect many SRGs with known parameters. It started here. This is about size, not quantity, and tries to give an idea how a typical SRG with certain parameters might look like. By now my collection has grown so big that I should advertise and describe it.

At the time of writing, my collections splits into four parts. The first two parts are graphs generated with GM- or WQH-switching. The second two parts are graphs generated with what I call Kantor switching. All the data is provided in graph6 format in a compressed text file. In total something beyond 190 million.

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Don’t be a Square (but count them)

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).)

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Sp(6, 2)’s Family, Plots, and Ramsey Numbers

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 Sp(6, 2), has parameters (63, 30, 13, 15). 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 Sp(6,2). Plots of the number of cliques, cocliques, and the size of the autmorphism group are scattered throughout this post.

size_aut

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