UVictoria Dynamics and Probability Seminar: Jon Noel

The homomorphism density of a graph H in a graph G is the probability that a random function from the vertex set of H to the vertex set of G is a graph homomorphism. A natural question studied by Leontovich and Sidorenko back in the 80s and 90s is: given two trees, say H and T, under what conditions does the homomorphism density function of H dominate the homomorphism density function of T? We apply a beautiful information-theoretic approach of Kopparty and Rossman to reduce this problem to solving a particular linear program. We then use this perspective to answer the question for various pairs H and T of trees. Roughly speaking, short bushy trees tend to be more numerous than tall skinny ones, but there are exceptions. Joint work with Natalie Behague, Gabriel Crudele and Lina M. Simbaqueba.