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.
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).)
A quick post about small Ramsey numbers. I like to write on my blog about things which I do not intend to publish, but also do not want to keep as private knowledge. This is one of these posts.
Stanisław P. Radziszowski writes the following in the 15th revision of his survey on small Ramsey numbers:
One can expect that the lower bounds in Table II are weaker than those in Table I, especially smaller ones, in the sense that some of them should not be that hard to improve, in contrast to the bounds in Table I.
Table II contains slightly larger Ramsey numbers, for instance and . So when one has idle CPU time, it seems to be reasonable to use it here, I thought some weeks ago. And indeed, it is. I found a witness for and a witness for for . This improves the previously known lower bounds by a gigantic 7 in the first case and a nearly as gigantic 6 in the second case. Here you can find a short description of both cases.
And if anyone asks: The upper bounds are much, much, much larger than both lower bounds. Also I did not put any effort into my search, so it is probably very feasible to improve a few more numbers in Table II of the survey.
I started writing this blog post some months ago. Occasion was that my paper “A construction for clique-free pseudorandom graphs” (in joint work Anurag Bishnoi and Valentina Pepe) was accepted by Combinatorica with minor revisions. More precisely, one of the referees was unfavorable of publication because he got the impressions that we are simply restating a result by Bannai, Hao and Song. I think that the referee had a point, but for slightly wrong reasons. This triggered me to do two things. First of all, it made me include more history of the construction in our actual paper. Then I wanted to write a blog post about the history of the construction. Sadly, I wanted to include too much history in my first attempt to write this post, so it was very much out of scope. Here now a more concise version of my original plan.
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.
Recently, I collected a short list of phrases for conjectures on a well-known social media platform and several people contributed to it. One can easily find more examples online, but I like my list, so I will keep it here and include references (as far as I have them). Probably, I will add more entries over time.
Firstly, I will give a list of phrases. Secondly, references for the phrases.
As Anurag Bishnoi likes to point out on his blog, an often overlooked source of wisdom is Willem Haemer’s PhD thesis from 1979. Many of Haemer’s proofs rely on simple properties of partitions of Hermitean matrices. My motivation for this post was a small exercise for myself. I wanted to prove the easy one of the two Cheeger inequalities for graphs using Haemer’s technique. Non-surprisingly, the book “Spectra of Graphs” by Brouwer and Haemers gives this kind of proof.
The K3xK3 graph with two highlighted equitable partitions. Think of the lines as cliques of size 3. This is a picture from my Master thesis many, many years ago.
Earlier on this blog we discussed that there are many different names for equitable partitions. My list from back then is incomplete as (1) I intentionally left out terms such as -designs (not to be confused with -designs) from Delsarte’s PhD thesis, and (2) I realized that for instance Stefan Steinerberger defined a notion of graphical designs which resembles the definition of a -design, while people working on latin squares call them -plexes, for instance see here. For this post just recall the following facts on equitable partitions: