Finding Graph Isomorphisms in Heated Spaces in Almost No Time

TL;DR

The graph isomorphism problem asks whether two graphs are identical up to relabeling, a task with known quasipolynomial-time algorithms but no general polynomial-time solution. The authors propose a curvature-based approach that extracts discrete curvature from the short-time heat-kernel expansion $K(t)=e^{-tL}$ of the graph Laplacian $L=D-A$, using these features to build candidate vertex correspondences and verify them to ensure one-sided correctness. Empirically, the method resolves a wide range of graph families, including highly symmetric cases, by combining spectral data with geometric augmentation (edge subdivisions and symmetry-breaking vertex attachments). While not a proof of polynomial-time GI, the results demonstrate that enriched spectral geometry can capture subtle structural distinctions invisible to eigenvalues alone and suggest a new research direction with practical implications and potential extensions to broader algorithmic problems.

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 symmetric 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 isomorphic 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 spectral 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.

Figures (17)

• Figure 1: The spectral dimension $d_s = 17$ is recovered for $\mathcal{G}_1$ with $N = 50$.
• Figure 2: The curvatures $u_{1,v}(\mathcal{G}_1)$ and $u_{1,v}(\mathcal{G}_2)$ are shown in the figures, with the vertical lines showing the indices of the mismatching nodes.
• Figure 3: There are 9 relabeled nodes, shown as the-off diagonal terms in the heatmap.
• Figure 4: The number of pairs below the dashed vertical lines denoting $C_{critical}$, which correspond to the gap in the sorted pairwise curvature difference, is $N$ confirming that $\mathcal{G}_1$ and $\mathcal{G}_2$ are isomorphic, while the curvature difference is shown in (b).
• Figure 5: Two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ are one edge-switch away from isomorphism. In Figure (a) the number of nodes below the first gap in $C_{critical}$ is not $N$ therefore the graphs are non-isomorphic. Further, the heatmap of the pairwise difference $u_{1,v}(\mathcal{G}_1)$ and $u_{1,v}(\mathcal{G}_2)$ for these random graphs of $N = 200$ and non-homogeneous degree is shown iin (b).