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Spectral Theory for Borel PMP Graphs

Cecelia Higgins, Pieter Spaas, Alexander Tenenbaum

Abstract

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence.

Spectral Theory for Borel PMP Graphs

Abstract

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by bounded-to-one functions is at most . Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence.
Paper Structure (18 sections, 58 theorems, 160 equations)

This paper contains 18 sections, 58 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Borel graphs and descriptive combinatorics
  4. Bounded operators on Hilbert spaces
  5. Mass transport for pmp graphs
  6. Operators associated to Borel graphs
  7. The adjacency operator
  8. The Laplacian operator
  9. First basic properties
  10. Spectral gap
  11. Bipartiteness
  12. Upper bounds on the chromatic number
  13. Wilf's theorem
  14. Graphs generated by functions
  15. Lower bounds on the chromatic number
  16. ...and 3 more sections

Key Result

Theorem A

Let $\mathcal{G}$ be a bounded-degree Borel pmp graph on a standard probability space.

Theorems & Definitions (128)

  • Theorem A: Theorems \ref{['thm:bipartiteimpliessymspec']} and \ref{['thm:-dspecimpliesbipartite']}
  • Theorem B: Theorem \ref{['thm: wilf pmp']}
  • Theorem C: Theorem \ref{['thm:nfunctions']}
  • Theorem D: Theorem \ref{['thm: hoffman bounded']}
  • Theorem E: Theorem \ref{['thm: brouwer haemers']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Corollary 2.4
  • ...and 118 more