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On the sum of the two largest eigenvalues of the curl-curl operator on graphs

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Pith's one-line read The sum of the two largest eigenvalues of curl^* curl is at most the sum of the first two terms in the conjugate of the second-order degree sequence.

desk verdict The paper proves the sum of the two largest eigenvalues of curl^* curl is bounded by the sum of the first two terms of the conjugate second-order degree sequence. read the letter →

arxiv 2606.26512 v1 pith:C2GDBR3O submitted 2026-06-25 math.CO

Yueli Han , Lu Lu , Jianfeng Wang This is my paper · ORCID
classification math.CO
keywords curl-curloperatormajorizationinequalitiessecond-orderdegreesequenceDuval-ReinerconjecturegraphHelmholtzianup-Laplaciansimplicialcomplexes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the first two majorization inequalities for the spectrum of the curl-curl operator on graphs, where triangles act as the 2-simplices. It shows that these eigenvalues are bounded above by the leading terms of the conjugate sequence that counts how many triangles meet at each vertex. The bound directly confirms the initial predictions of the Duval-Reiner conjecture in this setting and supplies concrete upper bounds for the two largest eigenvalues of the associated Helmholtzian. The same argument applies to the up-Laplacian of any 3-family, providing a partial advance on the conjecture in dimension one.

What carries the argument

The operator curl^* curl, whose eigenvalues are shown to satisfy the initial majorization relations with respect to the conjugate of the second-order degree sequence that records the number of incident triangles per vertex.

What would settle it

Explicit computation of the eigenvalues of curl^* curl and the conjugate second-order degree sequence on the complete graph K_4, checking whether their partial sums satisfy the stated inequality.

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Extended reading notes

Core claim

We prove that the sum of the two largest eigenvalues of curl^* curl does not exceed the sum of the first two entries of the conjugate of the second-order degree sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for curl^* curl. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian Δ1 = -grad div + curl^* curl. The same result extends to the up-Laplacian of any 3-family.

Load-bearing premise

The Duval-Reiner conjecture on graphs reduces to the claim that the spectrum of curl^* curl is majorized by the conjugate of the second-order degree sequence.

Editorial extensions

If this is right

  • The two largest eigenvalues of the graph Helmholtzian Δ1 are bounded above by the same conjugate terms.
  • The identical bound holds for the up-Laplacian on any 3-family of sets.
  • The result supplies the first two steps toward proving the full spectrum majorization for curl^* curl.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the full majorization holds, every eigenvalue of curl^* curl would be controlled by the same degree sequence, yielding bounds on all spectral gaps.
  • The technique may extend to higher-dimensional simplicial complexes by iterating the same conjugate-sequence comparison.
  • The bound can be checked directly on small graphs such as cycles with added triangles to test numerical sharpness.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the sum of the two largest eigenvalues of the curl^* curl operator on a graph (with triangles as 2-simplices) is at most the sum of the first two terms in the conjugate of the second-order degree sequence. This establishes the first two majorization inequalities predicted by the Duval-Reiner conjecture for this operator. Corollaries include upper bounds on the two largest eigenvalues of the graph Helmholtzian Δ_1 and an extension of the result to the up-Laplacian of any 3-family.

Significance. If the stated inequality holds, the result supplies a concrete partial confirmation of the simplicial analogue of the Grone-Merris theorem (proved by Bai in 2011), advancing the Duval-Reiner program by verifying the initial majorization steps for curl^* curl. The extension to 3-families and the derived bounds on the Helmholtzian are direct consequences that enlarge the scope of known spectral comparisons in higher-order graph operators.

minor comments (2)
  1. The abstract refers to 'the conjugate sequence' without an explicit definition or reference to its construction from the second-order degree sequence; a short inline definition or pointer to the relevant section would improve readability.
  2. Notation for the operators (curl^* curl, grad, div) is introduced without a preliminary section recalling the simplicial chain complex setup; adding a brief paragraph on the underlying 2-complex would clarify the setting for readers outside combinatorial topology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. The referee's description correctly identifies the main theorem and its place in the Duval-Reiner program. No specific major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper proves an inequality bounding the sum of the two largest eigenvalues of curl^* curl by the sum of the first two entries of the conjugate of the second-order degree sequence, using the definitions of the curl-curl operator, the variational characterization of eigenvalues, and the second-order degree sequence on graphs or 3-families. No step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from the authors' prior work; the result is a direct partial confirmation of the Duval-Reiner conjecture without load-bearing reliance on unverified internal assumptions or renaming of known results.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The argument rests on standard definitions and linear-algebra facts from algebraic topology and graph theory; no free parameters, ad-hoc axioms, or new entities are introduced.

assumptions (1)
  • standard math Standard spectral properties of the curl-curl operator and majorization on graphs with triangles as 2-simplices
    Invoked when reducing the Duval-Reiner conjecture to the curl^* curl spectrum.

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Pith. "Pith review of On the sum of the two largest eigenvalues of the curl-curl operator on graphs." pith.science (2026). https://pith.science/paper/C2GDBR3O

@misc{pith2026260626512,
  author       = {Pith},
  title        = {Pith review of: On the sum of the two largest eigenvalues of the curl-curl operator on graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2GDBR3O}},
  note         = {Machine review of arXiv:2606.26512}
}
abstract

The Grone--Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian $\Delta_0 = -\operatorname{div}\operatorname{grad}$ is majorized by the conjugate of the vertex degree sequence. Duval and Reiner proposed a simplicial complex analogue of this statement. On a graph, where triangles serve as $2$-simplices, their conjecture reduces to the assertion that the spectrum of $\operatorname{curl}^*\operatorname{curl}$ is majorized by the conjugate of the second-order degree sequence, which records the number of triangles containing each vertex. We prove that the sum of the two largest eigenvalues of $\operatorname{curl}^*\operatorname{curl}$ does not exceed the sum of the first two entries of that conjugate sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for $\operatorname{curl}^*\operatorname{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian $\Delta_1 = -\operatorname{grad}\operatorname{div} + \operatorname{curl}^*\operatorname{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval--Reiner conjecture in dimension~$1$.

Figures

Figures reproduced from arXiv: 2606.26512 by the authors.

Figure 1
Figure 1. The possibilities of G when D⊤ 2 (G) ≤ 2. First, we consider the case for D⊤ 2 (G) ≤ 2. If D⊤ 2 (G) = 0, then G ∼= t(G)K3. If D⊤ 2 (G) = 1, then G ∼= Fs ∪ (t(G) − s)K3 for some s ≥ 2, where Fs = K1 ∨ (sK2) is the friendship graph. If D⊤ 2 (G) = 2, then assume V2(G) = {u, v}. Therefore, we conclude that either G ∼= G1 when u 6∼ v, or G ∼= G2 when u ∼ v (shown in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. The graph G4 in Case 4. λ2 (curl∗ (G) curl(G)) = 3 ≤ D⊤ 2 (G), and (3) follows. Suppose now that the induction hypothesis holds for all graphs with fewer than t(G) triangles. Choose a triangle △ ∈ T(G) such that |△ ∩ V1(G)| ≥ 2. Let e be an edge of △ with both endpoints in V1(G), and let G0 = G − e. Then △ shares no edge with any other triangle. Therefore, one can easily verify that s(curl∗ (G) curl(G)) = s(curl∗ (G… view at source ↗
Figure 3
Figure 3. The structure of G (the edges in V2(G) are not shown). T1 = {△ ∈ T(G) | △ ∩ V1(G) 6= ∅}, and T2 = {△ ∈ T(G) | △ ∩ V1(G) = ∅}. Consider the 3-families T, T1 and T2. Then from T = T1 ⊔ T2 and the definition of ∂2∂ ∗ 2 (see [5]), it follows that ∂2(T)∂ ∗ 2 (T) = ∂2(T1)∂ ∗ 2 (T1) + ∂2(T2)∂ ∗ 2 (T2), (8) and that the ∂2(T)∂ ∗ 2 (T) = curl∗ (G) curl(G) . Therefore, combining Eq. (8) and Lemma 1, we get λ1 (curl∗ (G) curl(… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: G ∈ Ga1,...,a5 n;5 , with the adjacency relations among e1, . . . , e5 unspecified. (i) the three edges of each triangle ei ∪ {vi,j} are oriented according to the cyclic order; (ii) each such triangle inherits the orientation from the cyclic order in (i), as shown in …

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Duval--Reiner Conjecture: Counterexamples and the Second Partial-Sum Inequality

    math.CO 2026-07 accept novelty 8.0 of 10

    The Duval–Reiner majorization conjecture fails for every r≥5, while the r=2 inequality holds universally with an explicit equality classification.

Reference graph

Works this paper leans on

26 extracted references · cited by 1 Pith paper

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    F. W. Warner, Foundations of Differentiable Manifolds an d Lie Groups, Grad. Texts in Math. 94, Springer-Verlag, New York, 1983. A The eigenvalues of graphs in Fig. 1 For G ∈ { t(G)K3, F s ∪ (t(G) − s)K3, G 1}, no two triangles of G share an edge. Therefore, regardless of the c...

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  5. [13]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai3ai5(yx)ei3 (yx)ei5 + √ ai4ai5(yx)ei4 (yx)ei5 − √ ai3ai4(yx)ei3 (yx)ei4 . Suppose, to the contrary, that h(yx) < 0. If √ ai5 ⏐ ⏐(yx)ei5 ⏐ ⏐ = max j∈{ 3, 4, 5} √ aij ⏐ ⏐(yx)eij ⏐ ⏐, 23 set zx = |yx|. Otherwise, let t ∈ { 3, 4} ...

  6. [14]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai2ai3(yx)ei2 (yx)ei3 + √ ai3ai4(yx)ei3 (yx)ei4 +√ ai3ai5(yx)ei3 (yx)ei5 − √ ai4ai5(yx)ei4 (yx)ei5 . Suppose, to the contrary, that h(yx) < 0. If √ ai3 ⏐ ⏐(yx)ei3 ⏐ ⏐ = max j∈{ 3, 4, 5} √ aij ⏐ ⏐(yx)eij ⏐ ⏐, set zx := |yx|. Othe...

  7. [15]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, h(yx) = √ ai1ai3 (yx)ei1 (yx)ei3 + √ ai1ai2 (yx)ei1 (yx)ei2 + √ ai1ai4 (yx)ei1 (yx)ei4 − √ ai2ai3 (yx)ei2 (yx)ei3 − √ ai2ai4 (yx)ei2 (yx)ei4 − √ ai3ai4 (yx)ei3 (yx)ei4 . Suppose, to the contrary, that h(yx) < 0. If √ ai1 ⏐ ⏐(yx)ei1 ⏐ ⏐ = max j∈{ 1, 2, 3...

  8. [16]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai2ai3 (yx)ei2 (yx)ei3 + √ ai3ai4 (yx)ei3 (yx)ei4 + √ ai2ai4 (yx)ei2 (yx)ei4 + √ ai1ai4 (yx)ei1 (yx)ei4 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0. If √ ai4 ⏐ ⏐(yx)ei4 ⏐ ⏐ ≥ √ ai2 ⏐ ⏐(yx)ei2 ⏐ ⏐, then ...

  9. [17]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai1ai2 (yx)ei1 (yx)ei2 + √ ai1ai3 (yx)ei1 (yx)ei3 + √ ai4ai5 (yx)ei4 (yx)ei5 − √ ai2ai3 (yx)ei2 (yx)ei3 . Suppose, to the contrary, that h(yx) < 0. If √ ai1 ⏐ ⏐(yx)ei1 ⏐ ⏐ = max j∈{ 1, 2, 3} √ aij ⏐ ⏐(yx)eij ⏐ ⏐, set zx := |yx|....

  10. [18]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai2ai3 (yx)ei2 (yx)ei3 + √ ai1ai3 (yx)ei1 (yx)ei3 + √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai3ai4 (yx)ei3 (yx)ei4 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0. If √ ai3 ⏐ ⏐(yx)ei3 ⏐ ⏐ = max j∈{ 1, 2, 3} √ aij ⏐ ⏐(...

  11. [19]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai2ai4 (yx)ei2 (yx)ei4 + √ ai2ai3 (yx)ei2 (yx)ei3 + √ ai1ai4 (yx)ei1 (yx)ei4 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0. If √ ai4 ⏐ ⏐(yx)ei4 ⏐ ⏐ ̸= min j∈{ 1, 2, 4} √ aij ⏐ ⏐...

  12. [20]

    We claim that h(yx) ≥ 0

    Under this orientation, for any x > 0, h(yx) = √ ai2ai5 (yx)ei2 (yx)ei5 + √ ai2ai4 (yx)ei2 (yx)ei4 + √ ai2ai3 (yx)ei2 (yx)ei3 + √ ai1ai2 (yx)ei1 (yx)ei2 + √ ai1ai4 (yx)ei1 (yx)ei4 − √ ai1ai5 (yx)ei1 (yx)ei5 − √ ai3ai4 (yx)ei3 (yx)ei4 . We claim that h(yx) ≥ 0. Suppose, to the ...

  13. [21]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai2ai3 (yx)ei2 (yx)ei3 + √ ai3ai4 (yx)ei3 (yx)ei4 + √ ai1ai4 (yx)ei1 (yx)ei4 + √ ai1ai3 (yx)ei1 (yx)ei3 + √ ai2ai5 (yx)ei2 (yx)ei5 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0. If √ ai3 ⏐ ⏐(yx)ei3 ⏐ ⏐ ≥ ...

  14. [22]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai1ai5 (yx)ei1 (yx)ei5 + +√ ai1ai4 (yx)ei1 (yx)ei4 + √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai2ai5 (yx)ei2 (yx)ei5 + √ ai3ai4 (yx)ei3 (yx)ei4 − √ ai1ai3 (yx)ei1 (yx)ei3 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0...

  15. [23]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, h(yx) = √ ai1ai4 (yx)ei1 (yx)ei4 + √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai3ai5 (yx)ei3 (yx)ei5 + √ ai2ai5 (yx)ei2 (yx)ei5 + √ ai1ai5 (yx)ei1 (yx)ei5 − √ ai1ai3 (yx)ei1 (yx)ei3 − √ ai1ai2 (yx)ei1 (yx)ei2 − √ ai2ai3 (yx)ei2 (yx)ei3 . Suppose, to the contrary, tha...

  16. [24]

    33 Suppose, to the contrary, that h(yx) < 0

    Under this orientation, h(yx) = √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai3ai5 (yx)ei3 (yx)ei5 + √ ai2ai5 (yx)ei2 (yx)ei5 + √ ai1ai5 (yx)ei1 (yx)ei5 − √ ai3ai4 (yx)ei3 (yx)ei4 − √ ai2ai4 (yx)ei2 (yx)ei4 − √ ai1ai4 (yx)ei1 (yx)ei4 − √ ai1ai3 (yx)ei1 (yx)ei3 − √ ai1ai2 (yx)ei1 (yx)ei2 − √ ...

  17. [25]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, for any x > 0, h(yx) = √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai1ai5 (yx)ei1 (yx)ei5 + √ ai1ai3 (yx)ei1 (yx)ei3 + √ ai2ai5 (yx)ei2 (yx)ei5 + √ ai3ai4 (yx)ei3 (yx)ei4 − √ ai1ai2 (yx)ei1 (yx)ei2 . Suppose, to the contrary, that h(yx) < 0. If √ ai5 ⏐ ⏐(yx)ei5 ⏐ ⏐ ̸=...

  18. [26]

    Suppose, to the contrary, that h(yx) < 0

    Under this orientation, h(yx) = √ ai4ai5 (yx)ei4 (yx)ei5 + √ ai3ai4 (yx)ei3 (yx)ei4 + √ ai1ai4 (yx)ei1 (yx)ei4 + √ ai2ai4 (yx)ei2 (yx)ei4 − √ ai1ai3 (yx)ei1 (yx)ei3 − √ ai1ai2 (yx)ei1 (yx)ei2 − √ ai2ai3 (yx)ei2 (yx)ei3 . Suppose, to the contrary, that h(yx) < 0. If √ ai4 ⏐ ⏐(y...

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