Apparently, I described a very elegant argument to give a lower bound on the second largest eigenvalue of the adjacency matrix of a regular graph last year. This was pointed out in two recent preprints by Eero Räty, Benny Sudakov, and Istvan Tomon. This blog post is to describe the very short argument and how […]
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 […]
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 […]
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 […]
Let $ {X}$ be the orthogonality graph, that is the graph with $ {\{ -1, 1 \}^n}$ as vertices with two vertices adjacent if they are orthogonal. So $ {x, y \in \{ -1, 1 \}^n}$ are adjacent if $ {x \cdot y = x_1y_1 + x_2y_2 + \ldots + x_ny_n = 0}$. There are […]
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 $ {\{ 0, 1 \}^4}$ pairwise at Hamming distance at most $ {2}$ (a problem solved by Kleitman, recently investigated by Huang, Klurman and Pohoata). […]
Let us consider the $ {n}$-dimensional hypercube $ {\{ 0, 1 \}^n}$. The Hamming graph on $ {H_n}$ has the elements of $ {\{ 0, 1 \}^n}$ as vertices an two vertices are adjacent if their Hamming distance is one, so they differ in one coordinate. It is easy to see that the independence number […]
Last week I had a cold and could not do much thinking. So I spent my time making TikZ pictures for an upcoming talk of mine. This talk is on my recent work with Akihiro Munemasa on constructing cospectral strongly regular graphs. I think that the pictures are nice for a blog post, so here […]
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, […]
