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arxiv: 2605.15017 · v1 · submitted 2026-05-14 · 🧮 math.CO · math.OC· math.SP

Recognition: 2 theorem links

· Lean Theorem

Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries

Andrew Niu
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Pith reviewed 2026-05-15 03:04 UTC · model grok-4.3

classification 🧮 math.CO math.OCmath.SP MSC 05C50
keywords conformal rigiditygraph Laplaciansspectral embeddingssubdifferentialsorbit-isometriesvertex-transitive graphseigenvalue optimization
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The pith

A single eigenvector certifies conformal rigidity for vertex-transitive graphs and similar symmetric ones.

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

The paper develops a subdifferential framework that unifies variational optimization of Laplacian eigenvalues with the geometry of spectral embeddings. It introduces orbit-isometric embeddings, a symmetry-aware weakening of the edge-isometric condition, and proves this weaker notion still certifies whether uniform edge weights extremize the second-smallest or largest eigenvalue. For all vertex-transitive graphs and a broader class, the certification collapses to the existence of one suitable eigenvector. A reader cares because the method replaces numerical searches with algebraic checks that are exact and often reduce to linear feasibility.

Core claim

A graph is lower conformally rigid when uniform weights maximize lambda_2 and upper conformally rigid when they minimize lambda_n. The subdifferential analysis shows that an orbit-isometric embedding certifies either property, and symmetry reduction implies that for vertex-transitive graphs a single eigenvector suffices, yielding an algebraically exact test via linear feasibility or quadratic equations solved by Gröbner bases.

What carries the argument

The orbit-isometric spectral embedding, a symmetry-reduced analogue of edge-isometric embeddings that remains sufficient to certify conformal rigidity.

Load-bearing premise

That orbit-isometric embeddings characterize conformal rigidity just as edge-isometric ones do.

What would settle it

A vertex-transitive graph on which the single-eigenvector algebraic test declares rigidity but direct numerical maximization of lambda_2(w) or minimization of lambda_n(w) finds a non-uniform weight that improves the extremum.

Figures

Figures reproduced from arXiv: 2605.15017 by Andrew Niu.

Figure 1
Figure 1. Figure 1: The barbell graph with uniform edge weights (left) and a weight redistribution that strictly increases λ2 (right). This shows that the barbell graph is not lower conformally rigid. This example suggests that for a graph G to be lower conformally rigid, there cannot be any asymmetries in the density of the graph; otherwise we could shift weight from denser regions to sparser regions. Thus, for G to be lower… view at source ↗
Figure 2
Figure 2. Figure 2: The friendship graph F3 with uniform edge weights (left) and a weight redistribution that strictly decreases λn (right). This shows that F3 is not upper conformally rigid. As in the case of lower conformal rigidity, every edge of G must look the same to λn to be upper conformally rigid. Since λn measures the most oscillatory function on G, roughly speaking, we do not want hubs or asymmetries in the degrees… view at source ↗
Figure 3
Figure 3. Figure 3: A lower conformally rigid graph (A), an upper con￾formally rigid graph (B), and a lower and upper conformally rigid graph (C) (also known as the Desargues graph). 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cay(Z21, {1, 6}) and edge-isometric embeddings on Eλ2 and Eλn . To certify lower conformal rigidity, we must find eigenvectors φi ∈ Eλ2 such that (2) holds for all m = 42 edges. One can verify that λ2 = 4(1 − cos (2π/7)) with multiplicity 2. The following eigenvectors φ1 =  cos  2πj 7 20 j=0 , φ2 =  sin  2πj 7 20 j=0 . 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The map i 7→ i+ 7 (mod 21) is a graph automorphism of G = Cay(Z21, {1, 6}). Edge orbits are labeled by color. notion of an orbit-isometric spectral embedding as the symmetry reduced analogue of edge-isometric embeddings. Definition 2.6. We call a spectral embedding P orbit-isometric with respect to Ψ if there exists some c > 0 such that (4) 1 |Oi | X uv∈Oi ∥pu − pv∥ 2 = c for all edge orbits O1, . . . , Os… view at source ↗
Figure 6
Figure 6. Figure 6: Edge lengths of Cay(Z21, {1, 6}) under the orbit￾isometric embedding given by φ1  . for some c ∈ R. Instead of finding a collection of eigenvectors satisfying 42 con￾straints, we now just have to satisfy 2 constraints. Here, the single eigenvector φ1 = (cos (2πj/7))20 j=0 satisfies (3) with c = λ2/2. We illustrate the length of each edge in the corresponding spectral embedding P = [PITH_FULL_IMAGE:figure… view at source ↗
Figure 7
Figure 7. Figure 7: G = Cay(Z12, {2, 3}) with color-coded edge orbits. for k = 1, 4, 5. Since G is an abelian Cayley graph, we know that the set of orbit￾energy distributions ℓZ12 (Eλ2 ) = {ℓZ12 (φ) : φ ∈ Eλ2 } ⊆ R 2 is a convex polyhedral cone. We will later show that this cone has extreme rays generated by ℓZ12 (φ 1 1 ) = (6, 12), ℓZ12 (φ 4 1 ) = (18, 0), ℓZ12 (φ 5 1 ) = (6, 12). In particular, any orbit-energy distribution… view at source ↗
Figure 12
Figure 12. Figure 12: Lower conformal rigidity certificate to (12) for the graph on 8000 vertices. vertices, we achieved the runtimes shown in [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Valid (x0, x2) values for orbit-isometric embeddings of G on Eλ2 . Points indicate the embeddings used in [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The graph P25.01 (HoG 56682). The previous example illustrates that some conformally rigid graphs are degen￾erate in the sense that their relevant eigenspace Eλ is so large that they are still conformally rigid even in the absence of a large automorphism group. Intuitively, conformal rigidity is certified by finding a weighted collection of eigenvectors in 38 [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: ℓZ12 (Eλ2 ) for G = Cay(Z12, {2, 3}), with solution eigenvector φ = φ 1 1 + √ 1 3 φ 4 1 . The line c · (12, 12), c > 0 is in red. Example 2.7 (continued). We can now fully explain our exact certificate of lower conformal rigidity of G = Cay(Z12, {2, 3}). The group Z12 is abelian, so Theo￾rem 7.4 applies, and ℓΨ(Eλ2 ) is a polyhedral cone. The eigenspace Eλ2 = X1 ⊕X4 ⊕ X5 breaks into three 2-dimensional re… view at source ↗
Figure 16
Figure 16. Figure 16: Non-Cayley vertex-transitive (20, 10) graph: spring layout (left), PCA-projections of edge-isometric symmetrized em￾beddings on Eλ2 (center) and Eλn (right). Example 7.6. Let G be the non-Cayley, vertex-transitive graph (20, 10) on n = 20 vertices from Mathematica (HoG 56676). This graph was previously unable to be certified exactly using the techniques in [41]. Under the full automorphism group Ψ = Aut(G… view at source ↗
Figure 17
Figure 17. Figure 17: Exact rank-1 lower conformal rigidity certificate φφT to (24) for the graph on 8000 vertices. Even when Eλ does not decompose into distinct irreducibles, it is clear that Cone {ℓΨ(φj ) : 1 ≤ j ≤ h} is always a subset of ℓΨ(Eλ). Thus, we can always check this polyhedral cone for a solution. We are not guaranteed a solution this way, but in practice this can still quickly produce a certificate. There are si… view at source ↗
Figure 18
Figure 18. Figure 18: Symmetry reduction flowchart 7.3. Symmetry Reduction Pipeline. We now give a summary of our entire symmetry reduction pipeline shown in the flowchart shown in [PITH_FULL_IMAGE:figures/full_fig_p049_18.png] view at source ↗
read the original abstract

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $\lambda_n(w)$ over all normalized edge weights; $G$ is conformally rigid if it is lower or upper conformally rigid. This paper establishes a new framework for conformal rigidity through the language of subdifferentials, unifying the variational perspective on eigenvalue optimization with the geometry of edge-isometric spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gr\"{o}bner bases.

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

2 major / 2 minor

Summary. The paper introduces a subdifferential framework unifying variational optimization of Laplacian eigenvalues with the geometry of edge-isometric spectral embeddings to characterize conformal rigidity of graphs. It defines orbit-isometric embeddings as a symmetry-aware weakening of edge-isometry that remains sufficient for rigidity, then uses representation theory to show that for vertex-transitive graphs (and a larger class) rigidity is certified by a single eigenvector. This yields an algebraic certification procedure reducing in many cases to linear feasibility or Gröbner-basis solution of quadratic equations.

Significance. If the central claims hold, the work supplies an exact, non-numerical certification method for conformal rigidity that resolves an open question for vertex-transitive graphs and explains previously unexplained examples. The unification of subdifferentials, orbit-isometries, and representation-theoretic reduction constitutes a genuine technical advance with potential to streamline future work on symmetric graphs.

major comments (2)
  1. [Abstract / §3] Abstract and §3 (orbit-isometric definition): the reduction from full orbit-isometry on the eigenspace to a single-eigenvector check is load-bearing for the vertex-transitive claim. When multiplicity of λ₂ exceeds 1 (e.g., cycles C_n, n>3), the isometry condition must hold on the entire irreducible representation; the manuscript must supply an explicit invariance or projection argument showing why a single vector suffices, or the claim is not yet established.
  2. [§4] §4 (main theorem): the statement that orbit-isometric embeddings characterize conformal rigidity for the stated class relies on the subdifferential characterization of edge-isometric embeddings. The paper must verify that the subdifferential inclusion remains equivalent after the orbit-isometric relaxation; otherwise the sufficiency direction fails.
minor comments (2)
  1. [§2] Notation for the normalized weight set and the subdifferential operator should be introduced once with a dedicated display equation rather than inline.
  2. [§5] The Gröbner-basis certification procedure is mentioned only in the abstract; a brief complexity or termination remark in the main text would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of our work. The comments highlight important points that require clarification in the presentation. We address each major comment below and will revise the manuscript accordingly to strengthen the arguments.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and §3 (orbit-isometric definition): the reduction from full orbit-isometry on the eigenspace to a single-eigenvector check is load-bearing for the vertex-transitive claim. When multiplicity of λ₂ exceeds 1 (e.g., cycles C_n, n>3), the isometry condition must hold on the entire irreducible representation; the manuscript must supply an explicit invariance or projection argument showing why a single vector suffices, or the claim is not yet established.

    Authors: We appreciate this observation. In the manuscript, the orbit-isometric embedding is defined such that it respects the group action on the eigenspace. For vertex-transitive graphs, the relevant eigenspace forms an irreducible representation of the automorphism group. By the properties of irreducible representations and the fact that the condition is invariant under the group action, satisfaction for a single eigenvector (chosen generically) implies the condition for the entire space via projection onto the representation. However, to address the referee's concern directly, we will add an explicit lemma in Section 3 that provides the invariance argument: specifically, if the embedding is orbit-isometric for one vector in the eigenspace, then by averaging the inner products over the group orbits, the condition holds for all vectors in the space. This will make the reduction rigorous and explicit. revision: yes

  2. Referee: [§4] §4 (main theorem): the statement that orbit-isometric embeddings characterize conformal rigidity for the stated class relies on the subdifferential characterization of edge-isometric embeddings. The paper must verify that the subdifferential inclusion remains equivalent after the orbit-isometric relaxation; otherwise the sufficiency direction fails.

    Authors: We agree that the equivalence needs to be verified explicitly for the relaxed condition. In the proof of Theorem 4.1, the necessity direction follows directly from the definition, as orbit-isometry is weaker but still implies the edge conditions for the subdifferential. For sufficiency, we show that if the subdifferential condition holds for an orbit-isometric embedding, it corresponds to the variational characterization. To strengthen this, we will include a dedicated proposition in Section 4 that proves the subdifferential inclusion is preserved under the orbit-isometric relaxation by demonstrating that the supporting hyperplanes or subgradients align due to the symmetry reduction. This will confirm that the characterization remains valid. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent framework and certification

full rationale

The paper defines lower/upper conformal rigidity directly from the variational properties of Laplacian eigenvalues under normalized edge weights. It unifies this with the known (external) characterization via edge-isometric spectral embeddings and introduces the new, weaker orbit-isometric condition plus subdifferential language as sufficient for rigidity. For vertex-transitive graphs the single-eigenvector certification is derived via representation-theoretic reduction of the orbit condition; this step is presented as new content resolving an open question rather than a tautological renaming or self-citation reduction. No equations are shown to be equivalent by construction to fitted inputs, no ansatz is smuggled via self-citation, and the central claims retain independent mathematical content beyond the starting definitions. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard definitions of the Laplacian and normalized edge weights plus the new notions of orbit-isometry and subdifferential characterization; no free parameters are introduced in the abstract.

axioms (1)
  • domain assumption G is a connected undirected graph
    Stated at the opening of the definition of lower/upper conformal rigidity.
invented entities (1)
  • orbit-isometric embedding no independent evidence
    purpose: Weaker geometric condition than edge-isometry that still certifies conformal rigidity while respecting graph symmetries
    Newly defined in the paper to enable symmetry reduction and single-eigenvector certification.

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