Sidorenko-Type Inequalities for Pairs of Trees

TL;DR

This work introduces a Sidorenko-type partial order on graphs via homomorphism densities, focusing on trees. It develops a robust framework combining a forest-specialized linear program and an entropy-based analysis to certify inequalities $H\succcurlyeq T$, and applies it to classify all tree pairs on up to eight vertices. A complete star- and path-specific picture is obtained, including a precise criterion for $H\succcurlyeq S_k$ and the sharp $v(H)\ge4\iff H\succcurlyeq P_4$, with further necessary conditions (radius, degree sequence, independence) and a leaf-exploitation method. The results advance the understanding of Sidorenko-type inequalities, connect to graph limits via kernel formulations, and establish a rich repository of certificates and tables enabling exact poset determinations for small trees, with groundwork for larger classes and forest-generalizations.

Abstract

Given two non-empty graphs $H$ and $T$, write $H\succcurlyeq T$ 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 $H\succcurlyeq T$ 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 $H\succcurlyeq T$ for two forests $H$ and $T$ to solving a linear program of Kopparty and Rossman. We also characterize trees $H$ which satisfy $H\succcurlyeq S_k$ or $H\succcurlyeq P_4$, where $S_k$ is the $k$-vertex star and $P_4$ is the $4$-vertex path and resolve a problem of Csikvári and Lin.

Figures (5)

• Figure 1: A depiction of the homomorphisms $\varphi_1,\varphi_2$ and $\varphi_3$ from $P_6$ to $P_4$. Each vertex of $P_6$ is drawn directly below the vertex of $P_4$ that it is mapped to.
• Figure 2: A Hasse diagram for the poset of trees on at most $7$ vertices.
• Figure 3: The trees $T_1,\dots,T_{13}$. All non-empty trees on at most $6$ vertices, up to isomorphism.
• Figure 4: The trees $T_{14},\dots,T_{24}$. All trees on 7 vertices, up to isomorphism.
• Figure 5: The trees $T_{25},\dots,T_{47}$. All trees on 8 vertices up to isomorphism.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2: Sidorenko Sidorenko91
• Theorem 1.3: Godsil
• Theorem 1.4: Sağlam Saglam18
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: See Lovasz12
• proof : Proof of Theorem \ref{['th:poset']}