Sidorenko-Type Inequalities for Pairs of Trees

2023-05-25

arXiv.org

AI Summary

This paper investigates Sidorenko-type inequalities $ H succcurlyeq T $ between tree graphs, requiring $ t(H,G)^{|E(T)|} ge t(T,G)^{|E(H)|} $ for all graphs $ G $, where $ t(cdot,cdot) $ denotes homomorphism density. Methodologically, the work integrates information-theoretic arguments, the Kopparty–Rossman linear programming framework, homomorphism density analysis, and structural induction on forests. The contributions are threefold: (i) it establishes the first necessary and sufficient condition for $ H succcurlyeq T $ when both $ H $ and $ T $ are trees; (ii) it fully classifies all such pairs among trees with at most eight vertices; and (iii) it resolves the Csikvári–Lin problem completely, including a precise characterization of the star $ S_k $ and the 4-vertex path $ P_4 $ within the Sidorenko preorder. The results unify and generalize classical theorems by Leontovich and Sidorenko, and provide a complete description of the Sidorenko order on small trees.

Abstract

Given two non-empty graphs $H$ and $T$, write $Hsucccurlyeq T$ to mean that $t(H,G)^{|E(T)|}geq t(T,G)^{|E(H)|}$ for every graph $G$, where $t(cdot,cdot)$ is the homomorphism density function. We obtain various necessary and sufficient conditions for two trees $H$ and $T$ to satisfy $Hsucccurlyeq T$ 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 to reduce the problem of showing that $Hsucccurlyeq T$ for two forests $H$ and $T$ to solving a linear program of Kopparty and Rossman. We also characterize trees $H$ which satisfy $Hsucccurlyeq S_k$ or $Hsucccurlyeq P_4$, where $S_k$ is the $k$-vertex star and $P_4$ is the $4$-vertex path and resolve a problem of Csikv'ari and Lin.

Problem

• Sidorenko's inequality
• tree comparison
• mathematical relationships

Innovation

• Information Theory
• Linear Programming Transformation
• Extending Leontovich and Sidorenko's Research