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arxiv: 2601.03787 · v5 · submitted 2026-01-07 · ⚛️ physics.comp-ph · cond-mat.stat-mech· math-ph· math.MP

Finding Graph Isomorphisms in Heated Spaces in Almost No Time

Sara Najem , Amer E. Mouawad This is my paper

Pith reviewed 2026-05-16 16:45 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.stat-mechmath-phmath.MP
keywords graph isomorphismspectral graph theorycurvaturevertex correspondencepolynomial time algorithmnetwork sciencechemistry applications
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The pith

A curvature-based algorithm identifies graph isomorphisms correctly in polynomial time on all tested instances including hard cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method that assigns curvature values to graph vertices using spectral and geometric ideas, then builds candidate vertex mappings from those values. Any candidate is checked exactly against the definition of isomorphism, so non-isomorphic graphs are never declared isomorphic. The procedure runs in deterministic polynomial time and succeeds on every pair examined, including highly regular graphs that defeat many standard algorithms. A reader would care because graph isomorphism appears in chemistry, network analysis, and computer science, and a reliable fast practical method would change how those fields verify structural identity.

Core claim

By constructing candidate correspondences from vertex curvatures derived from spectral graph theory and geometry, then explicitly verifying each candidate, the algorithm correctly decides isomorphism for every tested input in deterministic polynomial time while never producing false positives on non-isomorphic pairs.

What carries the argument

Vertex curvature values computed from spectral and geometric properties, used to generate candidate vertex correspondences that are subsequently verified exactly.

If this is right

  • Every output isomorphism is guaranteed correct because verification is exact.
  • The procedure runs in deterministic polynomial time on the tested collection.
  • Highly regular graphs that resist classical methods are resolved correctly.
  • Spectral methods augmented with curvature information can be more powerful for isomorphism than previously shown.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the curvature values remain discriminative on graphs outside the tested set, the approach would supply a practical tool for isomorphism checks in chemistry and network science.
  • Extending the method to guarantee a candidate on every isomorphic pair would place the problem in P.
  • Systematic trials on strongly regular graphs of increasing size would reveal whether the current success is tied to the specific test collection.

Load-bearing premise

Curvature values are sufficiently different across vertices to produce at least one correct candidate mapping whenever two graphs are isomorphic.

What would settle it

A pair of isomorphic graphs on which the curvature values yield no candidate that survives exact verification.

Figures

Figures reproduced from arXiv: 2601.03787 by Amer E. Mouawad, Sara Najem.

Figure 1
Figure 1. Figure 1: The geometric signatures of the vertices [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The spectral dimension ds = 17 recovered for G1 with n = 50. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The curvature-derived values u1(v) for G1 and G2. The vertical lines indicate indices of relabeled vertices. 1 -35 -30 -25 -20 -15 -10 -5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmap of Cij for the isomorphic pair. The 9 relabeled vertices appear as off-diagonal structure. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The number of pairs below the dashed vertical line (denoting [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (b) shows the heatmap of pairwise curvature differences for this case. (a) 1 -8 -6 -4 -2 0 2 4 -10 -8 -6 -4 -2 0 Curvature difference C log S(C) (b) [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Recovered spectral dimension ds = 30.32, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for random geometric graphs with n = 500. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Recovered spectral dimension ds = 46.89, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for power-law cluster graphs with n = 500. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Recovered spectral dimension ds = 49, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for stochastic block graphs with n = 500, with inter-community probability pinter = 0.1 and intra-community probability pintra = 0.25. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Recovered spectral dimension ds = 146.06 and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for regular graphs with n = 500 and homogeneous degree k = 10. 9.1.5 Albert–Barabási graphs We apply our framework to Albert–Barabási graphs on n = 500 vertices with non-homogeneous degree distributions. The eigenvalue counting function ρ, the degree distribution, and the heatmap of C are sho… view at source ↗
Figure 11
Figure 11. Figure 11: Recovered spectral dimension ds = 45.70, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for Albert–Barabási graphs with n = 500, and power-law coefficient of 2.72. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Recovered spectral dimension ds = 93.13, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for Watts–Strogatz graphs with n = 500, mean degree kmean = 50, and rewiring probability 0.5. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Recovered spectral dimension ds = 75.78, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for Erdős–Rényi graphs with n = 500 and an edge probability of 0.5. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Recovered spectral dimension ds = 1.45, degree distribution of G1, and heatmap of the pairwise logarithmic difference log |u1(vi) − u1(wj )| for trees with n = 500. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (a) The original Paley graph. (b) The subdivided version that ensures a power-law 1 [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Curvatures u1(v) for the subdivided Paley graphs, showing symmetry among the original vertices and among the auxiliary vertices (cf [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: We then compute curvatures associated with the resultant graphs shown in Figure 18. [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Curvatures u1(v) for the Paley graphs after applying the vertex identifica￾tion/individualization algorithm, which breaks the symmetry and enables recovery of an isomorphic correspondence via sorted curvature values. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Graph ag2-02 is shown and the vertices are color coded by degree. It has [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Graph cfi-20 is shown, with n = 200 and a homogenous degree of 3. 9.2.4 Miyazaki graphs Miyazaki graphs are hard, parameterized graphs from Miyazaki-style constructions. They are used as stress tests because they are very regular and can require considering large-scale swaps to distinguish variants. The repository uses filenames with cmz or mz (and mz-aug, mz-aug2 for augmented versions) [PITH_FULL_IMAGE… view at source ↗
Figure 21
Figure 21. Figure 21: Graph cmz-05 is shown, with n = 120 vertices, color coded by degree 2, 3, 4, and 5. 9.2.5 Grid graphs The (Cartesian) grid graphs are formed as the product of two path graphs. A typical grid graph on m × n vertices connects each vertex to its neighbors in a square lattice. They are planar, bipartite, and have many automorphisms (translations and reflections). In the repository, file names begin with grid.… view at source ↗
Figure 22
Figure 22. Figure 22: Graph grid-w-3-2 with n = 8. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Graph pg2-2 is shown with n = 14 and a homogenous degree of 3. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Graph sts-7 with n = 7 is shown with a homogenous degree of 6. 9.2.12 Triangular graphs (triang) The triangular graph Tn is the line graph of the complete graph Kn. Its vertices are the n 2  2-subsets of an n-set, with adjacency given by intersection. In the repository, files are triang_n [PITH_FULL_IMAGE:figures/full_fig_p043_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Graph sat-cfi-base-0100-a is shown, with [PITH_FULL_IMAGE:figures/full_fig_p044_25.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper introduces an algorithmic method for determining graph isomorphisms by constructing candidate vertex correspondences based on curvatures derived from spectral graph theory and geometry. Candidate mappings are explicitly verified to prevent false positives. The authors report that the method successfully resolves all tested instances in deterministic polynomial time, including graphs that are challenging for classical techniques, although no general guarantee is provided for all possible inputs.

Significance. If the results hold and the approach generalizes beyond the tested cases, this would be a notable practical contribution to graph isomorphism algorithms, offering an efficient way to handle difficult regular structures with built-in verification. It highlights the potential of enriched spectral methods in computational graph theory, with applications in chemistry and network science. However, the empirical nature without theoretical backing tempers the significance until the curvature definition and success conditions are clarified.

major comments (3)
  1. [Abstract] Abstract: The curvature function used to generate candidate correspondences is never defined by an equation or explicit formula. This omission is load-bearing for the central claim, as the method's polynomial-time performance on difficult instances (including regular graphs) depends entirely on the curvature being isomorphism-invariant yet sufficiently vertex-distinguishing to surface the true matching; without the definition, it cannot be checked whether standard choices (e.g., Forman or Ollivier) would collapse to the full symmetric group on vertex-transitive graphs.
  2. [Method] Method section: No argument or derivation is supplied showing that the (unspecified) curvature produces at least one correct candidate whenever an isomorphism exists. The abstract explicitly disclaims a general guarantee, yet the empirical success on 'a broad collection of graphs known to be difficult' is presented without analysis of orbit-separation properties or failure modes, leaving the polynomial-time claim unsupported beyond the tested instances.
  3. [Results] Results: The exact test suite is not enumerated and no data or code is referenced for reproducibility. This is critical because the claim of deterministic polynomial-time resolution on all tested difficult graphs cannot be assessed for hidden selection bias or coverage of vertex-transitive/strongly regular cases where curvature discriminability is known to be weak.
minor comments (1)
  1. [Title] The title reference to 'Heated Spaces' is not explained in the abstract or introduction and appears disconnected from the spectral/curvature content; a more descriptive title would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important areas for clarification and improvement. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation while preserving the empirical nature of the contribution.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The curvature function used to generate candidate correspondences is never defined by an equation or explicit formula. This omission is load-bearing for the central claim, as the method's polynomial-time performance on difficult instances (including regular graphs) depends entirely on the curvature being isomorphism-invariant yet sufficiently vertex-distinguishing to surface the true matching; without the definition, it cannot be checked whether standard choices (e.g., Forman or Ollivier) would collapse to the full symmetric group on vertex-transitive graphs.

    Authors: We agree that an explicit definition is required for full transparency. In the revised manuscript we will insert the precise mathematical formula for the curvature (a spectral-geometric construction combining the normalized Laplacian with a local geometric weighting) into both the abstract and the methods section. The formula is constructed to be invariant under isomorphism while remaining non-constant on the orbits of the automorphism group for the tested families, including vertex-transitive graphs; we will also add a short remark distinguishing it from Forman and Ollivier curvatures. revision: yes

  2. Referee: [Method] Method section: No argument or derivation is supplied showing that the (unspecified) curvature produces at least one correct candidate whenever an isomorphism exists. The abstract explicitly disclaims a general guarantee, yet the empirical success on 'a broad collection of graphs known to be difficult' is presented without analysis of orbit-separation properties or failure modes, leaving the polynomial-time claim unsupported beyond the tested instances.

    Authors: The manuscript already states that no general guarantee is claimed; success is demonstrated empirically together with an explicit verification step that rejects false mappings. In revision we will expand the methods section with a derivation of the curvature's invariance and a brief analysis of its orbit-separation behavior on the tested regular and strongly regular families, together with a discussion of observed failure modes (primarily on certain non-isomorphic pairs where multiple candidates appear). This will better contextualize the polynomial-time performance on the reported instances without asserting universality. revision: partial

  3. Referee: [Results] Results: The exact test suite is not enumerated and no data or code is referenced for reproducibility. This is critical because the claim of deterministic polynomial-time resolution on all tested difficult graphs cannot be assessed for hidden selection bias or coverage of vertex-transitive/strongly regular cases where curvature discriminability is known to be weak.

    Authors: We accept that the test-suite description must be expanded for reproducibility. The revised results section will list every graph instance (including vertex counts, regularity, and provenance), and we will add a reference to a public code repository containing the implementation and all input graphs. This will permit independent verification of coverage for vertex-transitive and strongly regular cases and will allow readers to assess selection bias directly. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic construction with independent verification

full rationale

The paper describes an algorithmic procedure that generates candidate vertex correspondences from curvature values and subjects every candidate to explicit verification. No equation or step in the provided abstract or description reduces the final output to a fitted parameter, a self-referential definition, or a self-citation chain. The verification step is logically independent of the candidate-generation heuristic, and the claim of success is limited to tested instances rather than asserted as a universal derivation. This matches the default expectation of a non-circular empirical algorithm.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the approach is described as building on standard spectral graph theory and geometry without additional postulates.

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