Sidorenko-Type Inequalities for Pairs of Trees

05/25/2023

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by   Natalie Behague, et al.

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Given two non-empty graphs H and T, write H≽ T to mean that t(H,G)^|E(T)|≥ t(T,G)^|E(H)| for every graph G, where t(·,·) is the homomorphism density function. We obtain various necessary and sufficient conditions for two trees H and T to satisfy H≽ T and determine all such pairs on at most 8 vertices. This extends results of Leontovich and Sidorenko from the 1980s and 90s. Our approach applies an information-theoretic technique of Kopparty and Rossman to reduce the problem of showing that H≽ T for two forests H and T to solving a particular linear program. We also characterize trees H which satisfy H≽ S_k or H≽ P_4, where S_k is the k-vertex star and P_4 is the 4-vertex path.