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Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

Pierre de la Harpe

Abstract

The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this article is to review some examples of infinite graphs for which the spectral multiplicity function of the adjacency operator has been determined. The second purpose of this article is to show explicit examples of infinite connected graphs which are cospectral, i.e., which have unitarily equivalent adjacency operators, and explicit examples of infinite connected graphs which are uniquely determined by their spectrum.

Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

Abstract

The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this article is to review some examples of infinite graphs for which the spectral multiplicity function of the adjacency operator has been determined. The second purpose of this article is to show explicit examples of infinite connected graphs which are cospectral, i.e., which have unitarily equivalent adjacency operators, and explicit examples of infinite connected graphs which are uniquely determined by their spectrum.
Paper Structure (9 sections, 32 theorems, 74 equations)

This paper contains 9 sections, 32 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Spectral measures and the Hahn--Hellinger Multiplicity Theorem
  3. Spectrum, spectral measures, and dominant vectors
  4. Multiplication operators
  5. The Hahn--Hellinger Multiplicity Theorem, and spectral multiplicity functions
  6. The infinite ray, the infinite line, and the lattices
  7. Spherically symmetric infinite rooted trees
  8. Regular trees
  9. An uncountable family of cospectral graphs from GrNP--22

Key Result

Proposition 1.1

The adjacency operator of the infinite ray $R$ has spectrum $\mathopen[ -2, 2 \mathclose]$, scalar-valued spectral measure equivalent to Lebesgue measure, and uniform multiplicity one. The adjacency operator of the infinite line $L$ has spectrum $\mathopen[ -2, 2 \mathclose]$, scalar-valued spectral

Theorems & Definitions (57)

  • Proposition 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Proposition 2.1: existence and characterizations of dominant vectors for self-adjoint operators
  • proof : References for the proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 47 more