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.
Recently, while working on a research project, I got on a tangent. From this tangent, I got on another tangent and that is what I want to write about today: a very nice Boolean function. This example got rediscovered several times for different reasons and, as I try to emphasize from time, I believe that things that are getting rediscovered many times must be of particular importance.
So let us define our Boolean function. I will give three very similar definitions throughout this post, but I will start with only one. Put . Our Boolean function is defined as follows: Put (as ). For , write as with and . Then use this rule:
- If , then .
- If , then .
- If , then .
- If , then .
In the following, we list some properties of this function. Many of the concepts here are also discussed in an earlier post of mine.
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.
I spent the last few days in vain using several spectral arguments to bound the size of certain intersection problems. For instance what is the largest set of vectors in pairwise at Hamming distance at most (a problem solved by Kleitman, recently investigated by Huang, Klurman and Pohoata). Here the answer is and , , , , would be such an example. Or the largest set of vectors in pairwise at distance or . Here the answer is and an example is , , , , , , , . This is a problem which I recently investigated together with Hajime Tanaka, extending work by Frankl and others.
Usually, this playing around does not lead to anything. But this time …. It is actually the same. However, I did one useful thing which is the following: Generously counting, I do know five different easy spectral arguments which can be used to investigate these questions. This blog post presents these methods for the two problems mentioned above.
Permutation Groups, Analysis of Boolean Functions, Finite Geometry, Coding Theory and Algebraic Graph Theory
Important mathematical concepts get reinvented many times. In my recent work with Yuval Filmus we explored objects that are called (in random ordering) Boolean degree 1 functions, Cameron-Liebler line classes, equitable partitions, completely regular strength 0 codes with covering radius 1, intriguing sets, perfect colorings, sets with dual width 1, tactical decompositions or tight sets — all depending on the context and who you ask. While the article with Yuval explains to some extent how these notions connect, a research article does not seem to be the right format to explain all concepts in sufficient detail. This post tries to amend this. It also prepares a future post which will elaborate my research with Yuval in more detail. (more…)
Today’s topic combines three of my favorite subjects: Erdős-Ko-Rado theorems (EKR theorems), finite buildings and spectral techniques. All of these topics deserve their own books (and have them, here some examples which I read: Erdos-Ko-Rado Theorems: Algebraic Approaches by Chris Godsil and Karen Meagher, Spectra of Graphs by Andries E. Brouwer and Willem H. Haemers, The Structure of Spherical Buildings by Richard M. Weiss), so I will only touch these topics slightly.
My main aim is to present a variation of the EKR theorem which is motivated by questions about spherical buildings. The variation was recently formulated by Klaus Metsch, Bernhard Mühlherr, and me. If you already know spherical buildings, then you might prefer to read the introduction of our paper instead. (more…)