Graph properties, graph limits and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity.Comment: 24 page

Quasirandomness in hypergraphs

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.Comment: 19 page

More on quasi-random graphs, subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases. The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page

Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

Limits of kernel operators and the spectral regularity lemma

We study the spectral aspects of the graph limit theory. We give a description of graphon convergence in terms of converegnce of eigenvalues and eigenspaces. Along these lines we prove a spectral version of the strong regularity lemma. Using spectral methods we investigate group actions on graphons. As an application we show that the set of isometry invariant graphons on the sphere is closed in terms of graph convergence however the analogous statement does not hold for the circle. This fact is rooted in the representation theory of the orthogonal group

Cut distance identifying graphon parameters over weak* limits

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology. Dole\v{z}al and Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from subgraph densities, t(H,*). This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the "pumping property", and thus can be used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming graphs, and a new section 3.2 on index pumping in the regularity lemm