Finding Graph Isomorphisms in Heated Spaces in Almost No Time

TL;DR

This paper presents a novel spectral and geometric algorithm for graph isomorphism that efficiently distinguishes non-isomorphic graphs, especially challenging regular structures, with promising practical results.

Contribution

Introduces a spectral and geometric approach using curvature to construct and verify vertex correspondences, improving practical graph isomorphism testing.

Findings

Successfully resolves all tested difficult instances in polynomial time

Non-isomorphic graphs are never incorrectly identified as isomorphic

Enriched spectral methods outperform classical techniques in tested cases

Abstract

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging algorithmic task, particularly for highly regular structures. Here we introduce a new algorithmic approach based on ideas from spectral graph theory and geometry that constructs candidate correspondences between vertices using their curvatures. Any correspondence produced by the algorithm is explicitly verified, ensuring that non-isomorphic graphs are never incorrectly identified as isomorphic. Although the method does not yet guarantee success on all inputs, we find that it correctly resolves every instance tested in deterministic polynomial time, including a broad collection of graphs known to be difficult for classical techniques. These results demonstrate that enriched spectral methods can be far more powerful than previously understood, and suggest a promising direction for the practical resolution of the complexity of the graph isomorphism problem.