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Sums of Laplacian eigenvalues and sums of degrees

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Pith's one-line read The paper proves that in every simplicial complex, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian is at most the sum of the $(r+1)k$ largest $r$-degrees of codimension-one faces, a sharp bound that generalizes Ande

desk verdict New sharp degree-sum bound for Laplacian eigenvalues; solid and worth refereeing, but Lemma 2.8 is false as printed and needs correction. read the letter →

arxiv 2508.04209 v1 pith:UTOZ5BSE submitted 2025-08-06 math.CO

Alan Lew This is my paper · ORCID
classification math.CO MSC 05C5005E4515A18
keywords sumsofLaplacianeigenvaluessimplicialcomplexesupperdegreesequencesBrouwer'sconjectureDuval-ReinergraphKyFaninequality
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 proves a high-dimensional analogue of a classical graph inequality: for any simplicial complex $X$, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian is bounded by the sum of the $(r+1)k$ largest $r$-degrees of $(r-1)$-faces. The bound is sharp, and in the graph case $r=1$ it gives $\sum_{i=1}^k \lambda_i(L(G)) \le |E| + k^2$, improving earlier bounds for all $k\ge 3$ and moving toward Brouwer's conjecture. The same method verifies the Duval--Reiner conjecture for $(r+1)$-partite $r$-dimensional simplicial complexes. The overarching point is that degree sums, not just individual degree maxima, control Laplacian eigenvalue sums in every dimension.

What carries the argument

The proof's load-bearing object is a weighted $r$-th lower Laplacian $L' = L^-_r(X) - \sum_{i=1}^{(r+1)k} (1 - d/d^{(r)}_i(X)) L_i$, where $d$ is the $(r+1)k$-th largest $r$-degree and $L_i$ is the matrix supported on the $i$-th heaviest $(r-1)$-face, with exactly one nonzero eigenvalue $d^{(r)}_i(X)$. The weight function $w(\sigma)=\min\{d/\deg_X^{(r)}(\sigma),1\}$ makes every row of $L'$ have $\ell^1$-norm at most $(r+1)d$, so the Frobenius--Gershgorin bound gives $\lambda_1(L')\le (r+1)d$. Ky Fan's inequality then removes the $L_i$ terms one by one, leaving exactly the sum of the top $(r+1)k$ $r$-degrees. The fact that $L^+_{r-1}(X)$ and $L^-_r(X)$ share all nonzero eigenvalues (Corollary

What would settle it

Diagonalize $L^+_{r-1}(X)$ exactly for all simplicial complexes on a small vertex set and compare each $k$-sum with the corresponding $(r+1)k$ largest $r$-degree sum; the first violation would refute Theorem 1.6. A minimal check is a graph consisting of a perfect matching plus isolated vertices with $k$ chosen so $d_{2k}=0$, where the proof's weight $d/\deg$ is $0/0$ and a consistent convention must be supplied.

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Extended reading notes

Core claim

The central claim is Theorem 1.6: for any simplicial complex $X$, any $1\le r\le \dim(X)$, and any $k$ with $1\le k\le f_{r-1}(X)/(r+1)$, the sum of the $k$ largest eigenvalues of the $(r-1)$-th upper Laplacian $L^+_{r-1}(X)$ is at most the sum of the $(r+1)k$ largest $r$-degrees among $(r-1)$-faces. The inequality is tight: equality holds for complexes in which every $(r-1)$-face is contained in exactly one $r$-face, the graph case being a perfect matching. Anderson and Morley's bound $\lambda_1(L(G))\le d_1(G)+d_2(G)$ is exactly the case $r=1,k=1$. Two consequences are highlighted: the general bound $\sum_{i=1}^k \lambda_i(L^+_{r-1}(X)) \le f_r(X)+\binom{(r+1)k}{2}$, and, for graphs, $\sum

Load-bearing premise

The argument requires that every $(r-1)$-face can be assigned the weight $w(\sigma)=\min\{d/\deg_X^{(r)}(\sigma),1\}$; when $d=0$ and a face has $r$-degree $0$, the ratio is $0/0$, and the paper does not state the convention that such faces contribute zero, leaving a gap in the literal proof of Theorem 1.6.

Editorial extensions

If this is right

  • For every graph $G$ with $|V|\ge k$, $\sum_{i=1}^k \lambda_i(L(G))\le |E|+k^2$; this improves the previous best bounds for $k\ge 3$ and is the graph-level content of Theorem 1.8.
  • The general complex bound $\sum_{i=1}^k \lambda_i(L^+_{r-1}(X))\le f_r(X)+\binom{(r+1)k}{2}$ follows immediately, giving a weak form of the high-dimensional Brouwer bound proposed as Conjecture 1.5.
  • For $(r+1)$-partite $r$-dimensional complexes, the stronger partite decomposition verifies the Duval--Reiner conjecture for this class (Corollary 1.10).
  • The inequality is tight: complexes in which every $(r-1)$-face meets exactly one $r$-face achieve equality; for $r=1$ this is a perfect matching, and star forests are near-extremal.
  • Hereditary graph classes inherit the bound; in particular square-free graphs satisfy Brouwer's conjecture for all $k\ge 7$ and girth-at-least-5 graphs for all $k\ge 1$ (Proposition 5.3).

Reading between the lines

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

  • Extension: the same weighted-decomposition argument should transfer to signless Laplacians of arbitrary complexes; the paper states the analogue (Theorem 7.5) without proof, and the underlying row-sum and Ky Fan steps do not use signs.
  • Extension: Corollary 4.2 hints at an interpolation between Bai's conjugate-degree bound and the new degree-sum bound; optimizing the two averages may yield Brouwer's exact $|E|+\binom{k+1}{2}$ for all $k$, not just for the hereditary families treated in Section 5.
  • Extension: the equality case analysis suggests a structural question the paper leaves open: whether connected graphs can attain equality in Theorem 1.6 for $k>1$; if none can, the $|E|+k^2$ graph bound is never tight for connected graphs, strengthening the case for Brouwer's conjecture.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proves Theorem 1.6: for any simplicial complex X, 1 ≤ r ≤ dim X and 1 ≤ k ≤ f_{r-1}(X)/(r+1), the sum of the k largest eigenvalues of the upper Laplacian L^+_{r-1}(X) is bounded by the sum of the (r+1)k largest r-degrees of (r-1)-faces. This sharp inequality extends Anderson–Morley. Corollary 1.7 gives the general bound f_r(X)+binom((r+1)k,2); Theorem 1.8 gives the graph bound sum_{i=1}^k λ_i(L(G)) ≤ |E|+k^2, improving Theorem 1.3; Theorem 1.9 and Corollary 1.10 prove the Duval–Reiner conjecture for (r+1)-partite r-dimensional complexes. Section 5 gives applications to forests, bounded-degree, planar, square-free, girth, and path/cycle-free graphs; Section 7 discusses extensions to signless Laplacians. The proofs use elementary tools: Ky Fan's inequality, Gershgorin's theorem, boundary matrices, and Bai's theorem as an external input.

Significance. The main theorem, once corrected, is a substantial contribution: it unifies and generalizes Anderson–Morley, supplies the best-known general upper bound for sums of Laplacian eigenvalues, and the partite result resolves a natural special case of Duval–Reiner. The proof is self-contained and the extremal examples in Section 3.1 are informative. The paper is also honest about where it falls short of Brouwer's conjecture and about the signless analogue not following from Bai's theorem. However, the submitted text contains several load-bearing errors that must be fixed; they are local and repairable.

major comments (3)
  1. [Section 2.3, Lemma 2.8] Lemma 2.8 is false as printed. The correct identity is ε_k(G) = ε_{n−k−1}(\bar G) + nk − C(n,2), not with G on the right. The proof misapplies Lemma 2.7, which itself is misstated: it should read λ_i(L(\bar G)) = n − λ_{n−i}(L(G)). A counterexample to the printed lemma is G=K_4, k=1: ε_1(K_4)=−2, while the printed right-hand side gives ε_2(K_4)+4−C(4,2)=2+4−6=0. This lemma is used in Corollary 4.2's second case. The final bound can be recovered by applying the first case to \bar G, so the error is repairable, but the false statement must be corrected.
  2. [Section 4, Eq. (4.2)] In the proof of Theorem 4.1, the last sum on the right of (4.2) is '−∑_{i=2k+1}^n min{0,d_i−k}' but it must be '−∑_{i=2k+1}^n max{0,d_i−k}'. With the printed sign, adding (4.2) and (4.4) does not yield the claimed inequality for ε_k(G). This is a load-bearing typo for the derivation of Theorem 1.8.
  3. [Section 3, proof of Theorem 1.6] The weight w(σ)=min{d/deg_X(σ),1} is undefined when d=d^{(r)}_{(r+1)k}(X)=0 and deg_X(σ)=0. This case can occur: for example, r=1, k=2, and G=K_{1,2} has d=0. The proof should either handle d=0 separately (then the right-hand side equals the total degree sum and the inequality follows from trace) or state the convention that the corresponding L_i term is zero. As printed, Claim 3.3 and Claim 3.4 refer to an undefined expression. Claim 3.2's 'multiplicity one' assertion also fails for d_i=0, although its conclusion remains true. This is a gap in the main proof.
minor comments (3)
  1. [Section 2.3, Lemma 2.7] The statement as printed is missing the complement on the first Laplacian. It should read λ_i(L(\bar G)) = n − λ_{n−i}(L(G)) for 1 ≤ i ≤ n−1.
  2. [Section 3, after Eq. (3.2)] The sentence 'w(σ_i)=1 for (r+1)k ≤ i ≤ f_{r−1}(X)' should be 'for i > (r+1)k'; at i=(r+1)k the value is d/d_i, which is 1 only when d_i>0.
  3. [Throughout] There are several typographical issues, for example the displayed theorem statements in the abstract and main text are missing binomial coefficients due to formatting, and equation (4.2) should use max rather than min. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central Theorem 1.6 is derived from standard linear algebra; the only self-citation is a baseline being improved, not a load-bearing premise.

full rationale

I walked the derivation chain of Theorem 1.6. The proof uses standard tools: the AB/BA nonzero-eigenvalue coincidence (Lemma 2.5 / Corollary 2.6), Gershgorin's bound (Lemma 2.1), Ky Fan's inequality (Lemma 2.2), and a self-contained matrix construction (Lemma 3.1, Claims 3.2-3.4). The target sum of Laplacian eigenvalues is compared with a sum of degrees through an auxiliary matrix L' whose row sums are bounded; the argument does not assume the conclusion. No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the quantity being derived. Bai's theorem (Theorem 1.1) is an external result used only in the graphical application, not in the proof of Theorem 1.6; Anderson-Morley and Fan-Wu-Wang are also external. The only self-citation is the author's earlier paper [42], quoted as the baseline being improved in Theorem 1.3 and in the discussion of Theorem 1.8; it is never used as a proof ingredient. Sharpness is demonstrated by explicit examples (Proposition 3.5 and the star-forest examples), not by assuming the bound. I find no circular step. I do flag two non-circular correctness concerns that do not affect the circularity score: (1) Lemma 2.8 is false as printed: its proof applies Lemma 2.7 while dropping the complement, so the displayed identity ε_k(G)=ε_{n-k-1}(G)+nk-Δ(n,2) is invalid (e.g., K4, k=1); this affects the printed proof of Corollary 4.2, though the final bound may be recoverable by applying the first case to the complement. (2) In the proof of Theorem 1.6, w(σ)=min{d/deg(σ),1} is undefined when deg(σ)=0, and coefficients 1-d/d_i become 0/0 when d=0 (e.g., a triangle with isolated vertices and k=2); this is an omitted convention, fixable by setting w=0, and is not circularity. Omitted details in Theorem 7.5 are explicitly declared. Overall, the central claim is self-contained against external benchmarks and is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem is proved from standard linear-algebra facts plus one external theorem (Bai's proof of the Grone-Merris conjecture) in the application section. No free parameters are fitted, and no new entities are posited.

assumptions (5)
  • standard math Ky Fan inequality: for symmetric matrices A,B and 1<=k<=n, sum_{i=1}^k lambda_i(A+B) <= sum lambda_i(A) + sum lambda_i(B).
    Invoked as Lemma 2.2 in the main proof of Theorem 1.6, in Theorem 1.9 and in the signless analogue.
  • standard math Frobenius-Gershgorin bound: lambda_1(M) <= max row sum of absolute entries.
    Used as Lemma 2.1 to prove Claim 3.4, bounding the largest eigenvalue of L'.
  • standard math AB and BA have identical non-zero eigenvalues (Lemma 2.5).
    Used in Lemma 3.1 and Corollary 2.6 to connect upper and lower Laplacians.
  • domain assumption Bai's theorem (Grone-Merris conjecture): for graphs, sum of k largest Laplacian eigenvalues is at most the sum of the k largest conjugate degrees.
    Cited as Theorem 1.1 and used in the proof of Theorem 4.1 to derive the graph bound; not proved in this paper.
  • domain assumption Standard extremal edge bounds for graph families (Mantel, planar, square-free, girth >= 5, paths).
    Used in Proposition 5.3 within the hereditary-family framework; each is a known theorem.

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Pith. "Pith review of Sums of Laplacian eigenvalues and sums of degrees." pith.science (2026). https://pith.science/paper/UTOZ5BSE

@misc{pith2026250804209,
  author       = {Pith},
  title        = {Pith review of: Sums of Laplacian eigenvalues and sums of degrees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTOZ5BSE}},
  note         = {Machine review of arXiv:2508.04209}
}
abstract

Let $X$ be a simplicial complex. For $1\le i\le\dim(X)$, let $X(i)$ be the set of $i$-dimensional faces of $X$, and let $f_i(X)=|X(i)|$. For $0\le i\le \dim(X)-1$, let $L_i^+(X)$ be the $i$-th upper Laplacian operator of $X$. For $\sigma\in X$ and $1\le r\le \dim(X)$, we denote by $\text{deg}_X^{(r)}(\sigma)$ the number of $r$-dimensional faces of $X$ containing $\sigma$. For a symmetric matrix $M\in \mathbb{R}^{n\times n}$ and $1\le i\le n$, let $\lambda_i(M)$ be the $i$-th largest eigenvalue of $M$. We prove that for every complex $X$, $1\le r\le\dim(X)$, and $1\le k\le f_{r-1}(X)/(r+1)$, \[ \sum_{i=1}^k \lambda_i(L_{r-1}^+(X)) \le \max \left\{ \sum_{\sigma\in A} \text{deg}_X^{(r)}(\sigma) :\, A\subset X(r-1),\, |A|=(r+1)k \right\}. \] This bound is sharp, and it extends a classical result of Anderson and Morley, corresponding to the special case $k=1,\, r=1$. As a consequence, we show that for all $1\le r\le \dim(X)$ and $1\le k\le f_{r-1}(X)$, \[ \sum_{i=1}^{k} \lambda_i(L_{r-1}^+(X)) \le f_r(X) + \binom{(r+1)k}{2}. \] In the case $r=1$, we obtain the following improved bound: for every $k\ge 1$ and every graph $G=(V,E)$ with $|V|\ge k$, \[ \sum_{i=1}^k \lambda_i(L(G)) \leq |E|+k^2, \] where $L(G)=L_0^{+}(G)$ is the Laplacian matrix of $G$. This improves upon previously known bounds for all $k\ge 3$, and may be seen as a further step towards Brouwer's conjecture, which states that $\sum_{i=1}^k \lambda_i(L(G)) \leq |E|+\binom{k+1}{2}.$ As an additional application, we show that if $X$ is an $(r+1)$-partite $r$-dimensional simplicial complex on vertex set $V$, and $1\le k\le f_{r-1}(X)$, then \[ \sum_{i=1}^{k} \lambda_i(L_{r-1}^+(X)) \le \sum_{i=1}^k \left|\{v\in V:\, \text{deg}^{(r)}_X(v)\ge i\}\right|. \] This resolves a special case of a conjecture of Duval and Reiner, which states that the above inequality holds for all simplicial complexes.

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Forward citations

Cited by 5 Pith papers

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