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Maximising homomorphism counts between digraphs

Lukas Lüchtrath, Christian Mönch

Abstract

We prove a Sidorenko-type inequality for directed trees: for every oriented tree $T$ on $k$ vertices and every finite directed graph $G$, the homomorphism count hom$(T,G)$ is bounded above by the maximum of the two pure star counts hom$(S_{0,k-1},G)$ and hom$(S_{k-1,0},G)$. In other words, among all directed trees on $k$ vertices, the pure in- and out-stars maximise the homomorphism count into host digraphs. The proof is purely combinatorial, based on an iterative leaf-reallocation scheme combined with Hölder's inequality. We further investigate the corresponding homomorphism order on directed trees, discuss refinements via tail-truncation and pointwise bounds for rooted host graphs, and record several consequences, e.g. for random directed graph models and local weak limits, where the inequality reduces tree statistics to controlled pure in- and out-degree moments.

Maximising homomorphism counts between digraphs

Abstract

We prove a Sidorenko-type inequality for directed trees: for every oriented tree on vertices and every finite directed graph , the homomorphism count hom is bounded above by the maximum of the two pure star counts hom and hom. In other words, among all directed trees on vertices, the pure in- and out-stars maximise the homomorphism count into host digraphs. The proof is purely combinatorial, based on an iterative leaf-reallocation scheme combined with Hölder's inequality. We further investigate the corresponding homomorphism order on directed trees, discuss refinements via tail-truncation and pointwise bounds for rooted host graphs, and record several consequences, e.g. for random directed graph models and local weak limits, where the inequality reduces tree statistics to controlled pure in- and out-degree moments.
Paper Structure (18 sections, 19 theorems, 50 equations)

This paper contains 18 sections, 19 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Further contributions of the paper.
  3. Sidorenko's inequality for directed trees
  4. Directed stars
  5. Leaf-reallocation
  6. The homomorphism order on digraphs
  7. Extensions of Theorem \ref{['thm:main-directed-sidorenko']}
  8. Tail truncation
  9. Rooted message passing for pointwise bounds
  10. Applications
  11. From degree-moment convergence to LLNs for directed subgraph counts
  12. Further probabilistic applications
  13. Entropy maximisation and statistical models
  14. Matrix and walk inequalities
  15. Witnesses for universal incomparability in the homomorphism order between 3-arc directed trees
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $T$ be a directed tree on $k$ vertices. Then for every finite directed graph $H$, where $S_{0,k-1}$ and $S_{k-1,0}$ denote the pure in- and out-star graphs on $k$ vertices.

Theorems & Definitions (25)

  • Theorem 1.1: Directed Sidorenko inequality for trees
  • Lemma 2.1: Directed stars
  • Lemma 2.2: Leaf reallocation
  • Proposition 3.1: Complete incomparability for $3$-arc directed trees
  • Lemma 3.2: Incomparability of directed stars of given size
  • Remark 3.3: Discreteness conjecture and reversal-symmetrised order
  • Theorem 4.1: Directed tail inequality
  • Remark 4.2
  • Proposition 4.3: Rooted message passing identity
  • Proposition 4.4: Pointwise Hölder envelope for rooted homomorphism counts
  • ...and 15 more