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Graph isomorphism and multivariate graph spectrum

Wei Wang, Da Zhao

TL;DR

This work investigates the isomorphism problem through a spectral lens by connecting Weisfeiler--Leman indistinguishability to a generalized block Laplacian spectrum. It introduces a multivariate spectrum via $W_{m{A}}(m{s})$ and shows that identical multivariate spectra imply a fixed orthogonal conjugacy between associated matrices, underpinning a spectrum-based criterion that strengthens the link between $2$-WL and isomorphism. The main results show that $2$-WL indistinguishability ensures generalized block Laplacian spectral equivalence, and under certain arithmetic conditions on the last invariant factor of a Smith decomposition, that spectral equivalence forces graph isomorphism. Empirically, truncated and degree-based variants of the generalized block Laplacian spectrum demonstrate strong discriminative power on random graphs, supporting conjectures that almost all graphs are determined by these spectral data. This framework offers a scalable path to distinguishing graphs beyond traditional spectrum checks and provides new avenues for identifying isomorphism through algebraic invariants.

Abstract

We provide a criterion to distinguish two graphs which are indistinguishable by $2$-dimensional Weisfeiler-Lehman algorithm for almost all graphs. Haemers conjectured that almost all graphs are identified by their spectrum. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectrum.

Graph isomorphism and multivariate graph spectrum

TL;DR

This work investigates the isomorphism problem through a spectral lens by connecting Weisfeiler--Leman indistinguishability to a generalized block Laplacian spectrum. It introduces a multivariate spectrum via and shows that identical multivariate spectra imply a fixed orthogonal conjugacy between associated matrices, underpinning a spectrum-based criterion that strengthens the link between -WL and isomorphism. The main results show that -WL indistinguishability ensures generalized block Laplacian spectral equivalence, and under certain arithmetic conditions on the last invariant factor of a Smith decomposition, that spectral equivalence forces graph isomorphism. Empirically, truncated and degree-based variants of the generalized block Laplacian spectrum demonstrate strong discriminative power on random graphs, supporting conjectures that almost all graphs are determined by these spectral data. This framework offers a scalable path to distinguishing graphs beyond traditional spectrum checks and provides new avenues for identifying isomorphism through algebraic invariants.

Abstract

We provide a criterion to distinguish two graphs which are indistinguishable by -dimensional Weisfeiler-Lehman algorithm for almost all graphs. Haemers conjectured that almost all graphs are identified by their spectrum. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectrum.
Paper Structure (7 sections, 15 theorems, 56 equations, 1 figure, 2 tables)

This paper contains 7 sections, 15 theorems, 56 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notions and notations
  3. Main results
  4. Multivariate graph spectrum
  5. The Weisfeiler-Leman algorithm
  6. Proof of main theorems
  7. Discussion

Key Result

Theorem 1.1

Let $G$ and $H$ be two graphs of order $n$. Suppose the $2$-WL algorithm cannot distinguish $G$ and $H$, namely $G \cong_{2\text{-WL}} H$. Let $A$ and $B$ be the adjacency matrices of $G$ and $H$ respectively. Suppose the partition of vertices of $G$ by degree is given by $V = \bigsqcup_{i=1}^p V_i$

Figures (1)

  • Figure 1: Relation among equivalence classes of graphs

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: wang2016SquarefreeDiscriminantsMatrices
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 21 more