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Natural graph spectra

Ziqing Xiang

TL;DR

This work introduces natural graph spectra, defined as spectra of matrices obtained from the adjacency matrix by a fixed sequence of operations (linear combinations, matrix products, and Hadamard products), and develops a robust algebraic framework called double algebras. It proves a foundational sufficiency result: if the double-algebra generated by a graph’s adjacency matrix equals the full matrix algebra $M_n(\mathbb{F})$, then a corresponding natural spectrum determines the graph up to isomorphism, with random graphs satisfying this condition asymptotically almost surely. The key methodological advance is the universal $\circ$-idempotent basis within commutative, split semisimple $\circ$-subalgebras, enabling explicit construction and analysis of natural graph spectra, including a strong spectrum that aggregates information from many natural spectra. The findings show that a carefully designed spectral invariant can asymptotically determine random graphs, while also clarifying limitations in distinguishing certain graph families (e.g., distance-regular graphs) using natural spectra alone, and linking the framework to association schemes and finite algebraic structures. This offers a principled path to spectral GI-like strategies for random graphs and deepens the connection between spectral invariants and graph structure through double-algebra methods.

Abstract

In 2003, van Dam and Haemers posed a fundamental question in spectral graph theory: does there exist a ``sensible'' matrix whose spectrum determines a random graph up to isomorphism? This paper introduces the class of {\em natural graph matrices}, which are matrices defined by applying a fixed sequence of elementary operations to the adjacency matrix. This class includes many standard matrices such as the adjacency matrix, the Seidel matrix, the Laplacian matrix, and the distance matrix. We give an affirmative answer to the question of van Dam and Haemers by proving the existence of a natural graph matrix whose spectrum determines random graphs up to isomorphism. The proof introduces a new algebraic framework called {\em double algebras}, which provides a simple sufficient condition for spectral determination. This sufficient condition is then shown to hold for random graphs.

Natural graph spectra

TL;DR

This work introduces natural graph spectra, defined as spectra of matrices obtained from the adjacency matrix by a fixed sequence of operations (linear combinations, matrix products, and Hadamard products), and develops a robust algebraic framework called double algebras. It proves a foundational sufficiency result: if the double-algebra generated by a graph’s adjacency matrix equals the full matrix algebra , then a corresponding natural spectrum determines the graph up to isomorphism, with random graphs satisfying this condition asymptotically almost surely. The key methodological advance is the universal -idempotent basis within commutative, split semisimple -subalgebras, enabling explicit construction and analysis of natural graph spectra, including a strong spectrum that aggregates information from many natural spectra. The findings show that a carefully designed spectral invariant can asymptotically determine random graphs, while also clarifying limitations in distinguishing certain graph families (e.g., distance-regular graphs) using natural spectra alone, and linking the framework to association schemes and finite algebraic structures. This offers a principled path to spectral GI-like strategies for random graphs and deepens the connection between spectral invariants and graph structure through double-algebra methods.

Abstract

In 2003, van Dam and Haemers posed a fundamental question in spectral graph theory: does there exist a ``sensible'' matrix whose spectrum determines a random graph up to isomorphism? This paper introduces the class of {\em natural graph matrices}, which are matrices defined by applying a fixed sequence of elementary operations to the adjacency matrix. This class includes many standard matrices such as the adjacency matrix, the Seidel matrix, the Laplacian matrix, and the distance matrix. We give an affirmative answer to the question of van Dam and Haemers by proving the existence of a natural graph matrix whose spectrum determines random graphs up to isomorphism. The proof introduces a new algebraic framework called {\em double algebras}, which provides a simple sufficient condition for spectral determination. This sufficient condition is then shown to hold for random graphs.
Paper Structure (18 sections, 15 theorems, 35 equations)

This paper contains 18 sections, 15 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Natural graph matrices
  3. Double algebras
  4. Sufficient conditions
  5. Structure of the paper
  6. Double algebras
  7. Basics
  8. Free algebras
  9. Idempotents
  10. Spectrum and projection
  11. Universal idempotent basis
  12. Involution
  13. Natural graph spectrum
  14. The strong spectrum
  15. Graph of double subalgebras
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n$ be a natural number. There exists a natural graph spectrum (which may depend on $n$) such that for the Erdős-Rényi random graph $G(n, \frac{1}{2})$, the spectrum determines $G$ up to isomorphism asymptotically almost surely.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 24 more