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Principal eigenvectors and principal ratios in hypergraph Turán problems

Joshua Cooper, Dheer Noal Desai, Anurag Sahay

Abstract

For a general class of hypergraph Turán problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu, Cioabă--Gregory, and Zhang. The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues, and provide some preliminary evidence for our conjecture.

Principal eigenvectors and principal ratios in hypergraph Turán problems

Abstract

For a general class of hypergraph Turán problems with uniformity , we investigate the principal eigenvector for the -spectral radius (in the sense of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to ). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the -spectral radius of an arbitrary -graph in terms of the degrees of the graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu, Cioabă--Gregory, and Zhang. The case of our results leads to some subtleties connected to Nikiforov's notion of -tightness, arising from the Perron-Frobenius theory for the -spectral radius. We raise a conjecture about these issues, and provide some preliminary evidence for our conjecture.
Paper Structure (10 sections, 16 theorems, 61 equations, 1 figure)

This paper contains 10 sections, 16 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries and lemmata
  4. Spectral theory of hypergraphs
  5. Dense families and properties thereof
  6. Glossary for integer partitions
  7. Clonal families and related notions
  8. Upper bound on the principal ratio
  9. Proof of Theorem \ref{['thm: CTforpnorm']}
  10. Tight families and progress towards Conjecture \ref{['conj: bridgelessifftight']}

Key Result

Theorem 1.1

For $\mathcal{H}$ a set of $r$-graphs and $p > 1$,

Figures (1)

  • Figure 1: Illustration of the argument for Corollary \ref{['corr: 1bridgelessiff1tight']}.

Theorems & Definitions (60)

  • Definition 1.1: Keevash--Lenz--Mubayi
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.1: Nikiforov
  • Remark 1.5
  • Proposition 1.2
  • Remark 1.6
  • Definition 1.2
  • ...and 50 more