Weisfeiler-Leman Is Incomplete on Simple Spectrum Graphs, so Canonicalize Them

2026-05-22

AI Summary

Existing k-Weisfeiler-Leman (k-WL) tests fail to distinguish all non-isomorphic simple spectral graphs, thereby limiting the expressive power of graph neural networks (GNNs) on this class of graphs. This work proposes PRiSM, the first method to achieve provably complete canonical labeling for simple spectral graphs. PRiSM integrates spectral graph theory with a partition-refine-solve-match mechanism derived from eigendecomposition, and unifies it within a framework combining DeepSets and Transformer architectures. The approach not only offers theoretical guarantees of completeness but also achieves performance that significantly surpasses or matches current spectral canonicalization methods across graph classification, regression, and expressivity benchmarks. In doing so, PRiSM resolves a long-standing open problem concerning isomorphism testing for simple spectral graphs and the associated limitations in GNN expressiveness.

Abstract

Graphs with a simple spectrum admit cubic-time isomorphism testing, yet we prove that for every natural number $k$, the $k$-Weisfeiler-Leman ($k$-WL) test cannot distinguish all non-isomorphic graphs with a simple spectrum. As the WL hierarchy upper-bounds the distinguishing power of widely-used Graph Neural Networks (GNNs), this incompleteness applies to all such GNNs, ruling out completeness for every $k$-WL-aligned GNN family. To close this gap, we introduce PRiSM (Partition, Refine, Solve, Match), the first provably complete canonicalization of simple-spectrum eigendecompositions. PRiSM obtains the completeness guarantee that prior canonicalizations provably lack, and resolves the open problem of achieving complete expressivity on simple-spectrum graphs. When composed with DeepSets or a Transformer, PRiSM achieves universal approximation on simple-spectrum graphs, justifying the use of canonicalized Laplacian positional encodings. Empirically, PRiSM performs comparably to or outperforms existing spectral canonicalizations on graph regression, classification, and expressivity

Problem

• Weisfeiler-Leman
• simple spectrum
• graph isomorphism
• graph neural networks
• spectral canonicalization

Innovation

• Weisfeiler-Leman
• simple spectrum graphs
• graph canonicalization
• PRiSM
• graph neural networks