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On a Hierarchy of Spectral Invariants for Graphs

V. Arvind, Frank Fuhlbrück, Johannes Köbler, Oleg Verbitsky

TL;DR

The paper develops a hierarchical framework of graph invariants that augments eigenvalue information with angles between basis vectors and eigenspaces, capturing increasingly refined spectral data. By proving a tight walk-count characterization, it shows that invariants such as EA, weak-FSI, and strong-FSI are equivalent to specific levels of a walk-based hierarchy, and relates them to the Weisfeiler-Leman algorithms. The authors establish strict separations within the hierarchy (and from WL2), provide constructions based on strongly regular graphs to separate levels, and derive significant consequences, including that almost all graphs are determined by spectrum plus angles. This work clarifies the power and limits of spectral data for graph isomorphism testing and offers a combinatorial route to analyze complex spectral invariants through walk counts.

Abstract

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

On a Hierarchy of Spectral Invariants for Graphs

TL;DR

The paper develops a hierarchical framework of graph invariants that augments eigenvalue information with angles between basis vectors and eigenspaces, capturing increasingly refined spectral data. By proving a tight walk-count characterization, it shows that invariants such as EA, weak-FSI, and strong-FSI are equivalent to specific levels of a walk-based hierarchy, and relates them to the Weisfeiler-Leman algorithms. The authors establish strict separations within the hierarchy (and from WL2), provide constructions based on strongly regular graphs to separate levels, and derive significant consequences, including that almost all graphs are determined by spectrum plus angles. This work clarifies the power and limits of spectral data for graph isomorphism testing and offers a combinatorial route to analyze complex spectral invariants through walk counts.

Abstract

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.
Paper Structure (20 sections, 18 theorems, 80 equations, 2 figures)

This paper contains 20 sections, 18 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From isomorphism-invariant colorings to isomorphism invariants
  4. First two dimensions of combinatorial refinement
  5. A hierarchy of spectral invariants
  6. Characterization of the spectral invariants by walk counts
  7. Linear-algebraic background and known relations
  8. General characterization and its consequences
  9. Proof of Theorem \ref{['thm:char']}
  10. The Hierarchy
  11. Main eigenvalues and angles
  12. The generalized spectrum
  13. Separations
  14. $\mathsf{WL}_{3/2}$ is not stronger than $\mathop{\mathrm{\mathsf{strong-FSI}}}\nolimits$
  15. $w_*^{(2)}$ is not stronger than $\mathsf{WL}_{1}$
  16. ...and 5 more sections

Key Result

Lemma 2

$P_*^{(r)}\equiv\alpha_*^{(r)}$ for every integer $r\ge0$.

Figures (2)

  • Figure 1: Relations between graph invariants. An arrow $\mathcal{I}\to\mathcal{I}'$ means $\mathcal{I}'\preceq\mathcal{I}$.
  • Figure 2: Construction of $G(A',B')$.

Theorems & Definitions (31)

  • Lemma 2
  • proof
  • Lemma 3: folklore; see, e.g., GarijoGN11
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Corollary 7
  • proof
  • Corollary 8
  • proof
  • ...and 21 more