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On the satisfaction frequency of spectral characterization conditions

Nikita Lvov, Alexander Van Werde

Abstract

We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial.

On the satisfaction frequency of spectral characterization conditions

Abstract

We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial.

Paper Structure

This paper contains 27 sections, 29 theorems, 108 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Specific predictions for satisfaction frequencies
  3. Conditions based on the walk matrix
  4. Conditions based on the discriminant
  5. Two-step approach based on abstract-algebraic random matrix statistics
  6. On universality
  7. Outline
  8. Background and notation
  9. Definition and interpretation of cokernels
  10. Profinite completion and its Haar distribution
  11. On the special cases Z and Z[x]
  12. Pieces of modules and relevant properties
  13. Rephrasing conditions using the Z[x]-module structure of coker(M-xI)
  14. Regarding the walk matrix
  15. Regarding the discriminant
  16. ...and 12 more sections

Key Result

Theorem 1.3

Consider $\zeta\in \mathbb{Z}^n$ and $\mathbf{M}\in \mathbb{Z}^{n\times n}$ with $\mathbf{M} = \mathbf{M}^{\top}$ and suppose that $\det(\mathbf{W})$ is square-free. Then, $(\mathbf{M},\zeta)$ is characterized by $\Phi$-spectrum up to signed permutation.

Theorems & Definitions (71)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Van Werde van2025Sufficient
  • Conjecture 1.4
  • Definition 1.5
  • Theorem 1.6: Wang and Yu wang2016square
  • Conjecture 1.7
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 61 more