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A lower bound on the spectrum of unimodular networks

Mustazee Rahman

TL;DR

An Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs is proved, which generalizes the Alon Boppana Theorem for regular graphs.

Abstract

Unimodular networks are a generalization of finite graphs in a stochastic sense. We prove a lower bound to the spectral radius of the adjacency operator and of the Markov operator of an unimodular network in terms of its average degree. This allows to prove an Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs, which generalizes the Alon-Boppana theorem for regular graphs. A key step is establishing a lower bound to the spectral radius of a unimodular tree in terms of its average degree. Similarly, we provide a lower bound on the volume growth rate of an unimodular tree in terms of its average degree.

A lower bound on the spectrum of unimodular networks

TL;DR

An Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs is proved, which generalizes the Alon Boppana Theorem for regular graphs.

Abstract

Unimodular networks are a generalization of finite graphs in a stochastic sense. We prove a lower bound to the spectral radius of the adjacency operator and of the Markov operator of an unimodular network in terms of its average degree. This allows to prove an Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs, which generalizes the Alon-Boppana theorem for regular graphs. A key step is establishing a lower bound to the spectral radius of a unimodular tree in terms of its average degree. Similarly, we provide a lower bound on the volume growth rate of an unimodular tree in terms of its average degree.

Paper Structure

This paper contains 21 sections, 11 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Unimodular network and mass transport principle
  3. Examples.
  4. Local convergence.
  5. Spectral radius.
  6. Universal cover.
  7. Statement of results
  8. Outline of the paper
  9. Preliminaries
  10. Spectrum of a unimodular network
  11. Edge rooted graphs and non-backtracking walk
  12. Entropy
  13. Entropy of the NBW step.
  14. Spectral radius of unimodular trees
  15. Proof of Proposition \ref{['prop:1']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(T,\circ)$ be a unimodular tree with $\mathbf{E}\left [ \mathrm{deg}(\circ) \right ] < \infty$ and no leaves almost surely. Then Additionally, if $(T,\circ)$ has deterministically bounded degree then

Figures (1)

  • Figure 1: A 6-step height profile and a closed walk on the tree associated to it. Steps 3 and 4 each have two possible choices for a forward step. The stack $S$ updates as $[1] \to [\,] \to [3] \to [3,4] \to [3] \to [\,]$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1: Stationarity of NBW
  • proof
  • Proposition 1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • ...and 7 more