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

An approximate version of Sidorenko's conjecture

A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page