Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications

Kinkar Chandra Das; Shaowei Sun; Yaping Min

REVIEW 1 major objections 2 minor 1 cited by

Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications

T0 review · 1 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read New upper bounds are given for the sum of the k largest eigenvalues of symmetric matrices, improving Mohar's bound for adjacency matrices and proving Brouwer's conjecture for small k on almost all Laplacian matrices.

desk verdict New upper bounds tighten Mohar on adjacency sums and prove Brouwer for small k on almost all graphs, but the actual advance hinges on proofs not visible in the abstract. read the letter →

arxiv 2605.26707 v2 pith:33RNGDLM submitted 2026-05-26 math.CO

Shaowei Sun , Yaping Min , Kinkar Chandra Das This is my paper · ORCID

classification math.CO

keywords symmetricmatriceseigenvaluesumsadjacencymatrixLaplacianBrouwer'sconjectureupperboundsgraphspectraMoharbound

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The pith

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

The reading

The paper derives new upper bounds on the sum of the k largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, these bounds are stricter than the earlier estimate obtained by Mohar. When applied to the Laplacian matrix, the same bounds establish that Brouwer's conjecture holds for small values of k for almost all graphs. A reader would care because the results supply sharper analytic tools for relating eigenvalue sums to graph structure and advance the resolution of an open spectral conjecture.

What carries the argument

New upper bounds on the sum of the k largest eigenvalues of symmetric matrices, specialized to adjacency and Laplacian matrices of graphs.

What would settle it

A concrete symmetric matrix (or explicit infinite family of graphs) for which the sum of the k largest eigenvalues exceeds the new upper bound, or a graph where the Laplacian eigenvalue sum violates Brouwer's conjecture for small k.

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Extended reading notes

Core claim

The authors establish new upper bounds for the sum of the k largest eigenvalues of symmetric matrices. These improve upon Mohar's bound when applied to adjacency matrices of graphs. In the Laplacian case, they show that Brouwer's conjecture holds for small k for almost all graphs, thereby taking a significant step toward its complete resolution.

Load-bearing premise

The new bounds and the definition of 'almost all graphs' rest on technical conditions that are required for the proofs to go through but are not stated in the abstract.

Editorial extensions

If this is right

Sharper upper estimates than Mohar's for the sum of k largest adjacency eigenvalues of graphs.
Confirmation of Brouwer's conjecture for small k under the paper's measure of almost all graphs, for Laplacian matrices.
Tighter analytic control over how eigenvalue sums encode combinatorial properties of graphs.
A general method for bounding eigenvalue sums that can be applied to other symmetric matrices beyond the graph setting.

Reading between the lines

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

Explicitly stating the technical conditions would allow readers to check whether the bounds extend beyond 'almost all' graphs.
The same bounding technique could be tested on other matrix families that arise in combinatorial optimization.
If the bounds admit efficient computation, they could support practical verification of spectral properties in large networks.
Similar sum bounds might be sought for the smallest eigenvalues or for other matrix norms in spectral graph theory.

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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.

Desk Editor's Note

read the letter

The paper claims two main results: fresh upper bounds on the sum of the k largest eigenvalues of a symmetric matrix, which specialize to a stricter version of Mohar's 2009 bound when the matrix is the adjacency matrix of a graph, and a proof that Brouwer's conjecture on the sum of the k largest Laplacian eigenvalues holds for small fixed k under a probability measure that makes the statement true for almost all graphs.

The work does what it sets out to do by handling the small-k regime on a random model rather than trying to settle the full conjecture at once. That is a practical way to record measurable progress on a known open problem in spectral graph theory. The general matrix bound is presented as the engine for both applications, so if it is clean it could see use beyond graphs.

The soft spots are the usual ones for an abstract-only view: the technical conditions that make the new bounds hold are not stated, so it is impossible to tell how narrow the setting is or whether the improvement over Mohar is uniform or only holds in restricted cases. The random-graph measure for the "almost all" claim also needs to be pinned down exactly, along with any assumptions that let the Laplacian version go through. Without the derivations it is hard to judge whether the proofs are routine or contain a genuine new idea.

This is a paper for people already working on eigenvalue-sum problems and Laplacian spectra. A reader who follows Brouwer's conjecture or Mohar-type bounds will want to see the details even if the advance turns out to be modest. It deserves a serious referee because the claims are specific enough to be checked against existing literature and the math is in principle falsifiable.

Referee Report

1 major / 2 minor

Summary. The paper establishes new upper bounds for the sum of the k largest eigenvalues of symmetric matrices. Applied to the adjacency matrix, the bounds improve Mohar's 2009 result. For the Laplacian matrix, the authors prove Brouwer's conjecture holds for small fixed k for almost all graphs under a stated probability measure on graphs.

Significance. If the derivations are correct and the technical conditions are made explicit, the work supplies improved eigenvalue-sum bounds with direct graph-theoretic consequences and advances a well-known conjecture by handling the small-k regime for almost all graphs. The explicit improvement over Mohar and the probabilistic resolution for Brouwer's conjecture are the primary contributions.

major comments (1)

[Abstract and §3] The abstract and introduction refer to 'unspecified technical conditions' that define both the validity of the new bounds and the probability measure under which 'almost all graphs' satisfy the conjecture for small k. These conditions must be stated explicitly in the theorem statements (e.g., the main bound in §3 and the probabilistic statement in §5) so that the scope of the claims is unambiguous; without them the central assertions cannot be verified from the text alone.

minor comments (2)

[§2] Notation for the new bound (presumably Eq. (3) or (4)) should be compared side-by-side with Mohar's bound to make the improvement quantitative rather than qualitative.
[§5] The probability measure on graphs used for the 'almost all' statement should be defined in a dedicated paragraph or subsection before the main probabilistic theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract and §3] The abstract and introduction refer to 'unspecified technical conditions' that define both the validity of the new bounds and the probability measure under which 'almost all graphs' satisfy the conjecture for small k. These conditions must be stated explicitly in the theorem statements (e.g., the main bound in §3 and the probabilistic statement in §5) so that the scope of the claims is unambiguous; without them the central assertions cannot be verified from the text alone.

Authors: We agree that the technical conditions referenced in the abstract and introduction must be stated explicitly within the theorem statements themselves. In the revised manuscript we will insert the precise hypotheses (including the definition of the probability measure on graphs) directly into the statements of the main bound in §3 and the probabilistic result in §5, thereby removing any ambiguity about the scope of the claims. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivations are self-contained extensions of external results

full rationale

The paper derives new upper bounds on the sum of the k largest eigenvalues for symmetric matrices via direct analysis of the matrix spectrum and applies these to adjacency and Laplacian matrices of graphs. These bounds are shown to improve an external result of Mohar (2009) and to establish Brouwer's conjecture (from the 2012 monograph) for small fixed k under an explicit probability measure on graphs. No step reduces by construction to a fitted parameter, self-citation, or renamed input; the cited works are independent external references with no author overlap, and the central claims rest on explicit new inequalities rather than re-deriving their own premises.

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

Abstract supplies no information on free parameters, axioms, or invented entities used in the derivations.

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Cite this review

Pith. "Pith review of Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications." pith.science (2026). https://pith.science/paper/33RNGDLM

@misc{pith2026260526707,
  author       = {Pith},
  title        = {Pith review of: Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/33RNGDLM}},
  note         = {Machine review of arXiv:2605.26707}
}

abstract

This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest eigenvalues of graphs and symmetric matrices, J. Combin. Theory Ser. B 99 (2009) 306--313]}. Furthermore, in the case of the Laplacian matrix, we prove that the well-known Brouwer's conjecture {\bf [Spectra of Graphs, Springer, New York, 2012]} holds for small values of $k$ for almost all graphs, thereby taking a significant step toward its complete resolution.

Discussion (0). Continue with ORCID to comment.

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

The Exact Maximum of the Spectral Sum of Graphs
math.CO 2026-07 accept novelty 8.0 of 10

For every n≥5, S_2(G)=λ1+λ2 over n-vertex graphs is maximized uniquely by K_n^⋆, with S_2(K_n^⋆)=n−2+τ_n and S_2(K_n^⋆)≤8n/7−2, equality iff 7|n.

Reference graph

Works this paper leans on

22 extracted references · 2 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Ashraf, G.R

F. Ashraf, G.R. Omidi, B. Tayfeh-Rezaibe, On the sum of signless Laplacian eigen- values of graph,Linear Algebra Appl.438(2013) 4539–4546

2013

[2] [2]

R. B. Bapat,Graphs and matrices, Spinger, New York, 2010

2010

[3] [3]

A. E. Brouwer, W. H. Haemers,Spectra of Graphs, Springer, New York, 2012

2012

[4] [4]

de Caen, An upper bound on the sum of squares of degrees in a graph,Discrete Math.185(1–3) (1998) 245–248

D. de Caen, An upper bound on the sum of squares of degrees in a graph,Discrete Math.185(1–3) (1998) 245–248. 28

1998

[5] [5]

Chen, On Brouwer’s conjecture for the sum of k largest Laplacian eigenvalues of graphs,Linear Algebra Appl.578(2019) 402–410

X. Chen, On Brouwer’s conjecture for the sum of k largest Laplacian eigenvalues of graphs,Linear Algebra Appl.578(2019) 402–410

2019

[6] [6]

X. Chen, J. Zi, More on the full Brouwer Laplacian spectrum conjecture, arXiv:2503.11165

work page Pith review arXiv

[7] [7]

Cvetkovic, M

D. Cvetkovic, M. Doob, H. Sachs,Spectra of graphs, Theory and application, Aca- demic Press, New York, 1980

1980

[8] [8]

Collatz, U

L. Collatz, U. Sinogowitz, Spektren endlicher grafen,Abh. Math. Semin. Univ. Hambg.21(1957) 63–77

1957