Pith. sign in

REVIEW 4 minor 24 references

The positive and negative square-energy conjecture

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that every connected graph on n vertices satisfies min{s+(G), s−(G)} ≥ n−1, confirming the 2016 square-energy conjecture and several related spectral bounds.

desk verdict This paper genuinely proves the 2016 square-energy conjecture with a self-contained inductive argument, and the delicate averaging step checks out. read the letter →

arxiv 2607.18031 v1 pith:MJXZDAO7 submitted 2026-07-20 math.CO

Yinchen Liu , Quanyu Tang , Shengtong Zhang This is my paper · ORCID
classification math.CO MSC 05C5015A1815A42
keywords positivesquareenergynegativeadjacencyeigenvaluesdoublynonnegativematrixspectralgraphtheorysemidefiniteprogrammingp-energyNordhaus–Gaddumbounds
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

The paper proves a conjecture that has remained open since 2016: in any connected graph on n vertices, the smaller of the sums of squares of the positive and negative adjacency eigenvalues is at least n−1. The proof is built around a new inequality that holds for every doubly nonnegative matrix M attached to a connected graph: four times the square of the sum of square roots of edge entries is at most (2|E|−n+1) times the total mass of M. Applying the inequality to the entrywise squares of the positive and negative spectral parts of the adjacency matrix yields s±(G) ≤ 2|E|−n+1, which combined with s+ + s− = 2|E| forces the lower bound. The same result settles the disconnected version, the squared-spread bound, the adjacency-inertia conjecture, positive and negative p-energy conjectures, and a Nordhaus–Gaddum-type complement inequality.

What carries the argument

The load-bearing object is Theorem 2.1, a doubly-nonnegative inequality: 4(Σ_{edges} √M_uv)^2 ≤ q(G)·1^TM1, where q(G)=2|E|−n+1. It works by relaxing the exact spectral objects—Hadamard squares A±∘A±, which are positive semidefinite and entrywise nonnegative—to the full cone of doubly nonnegative matrices, a larger and more flexible class. The proof drives the inequality by induction: the cut-vertex case uses a Gram-vector decomposition so both the edge sum and the total mass split along the cut; the no-cut-vertex case averages the induction hypothesis over all one-vertex deletions, where the identities Σ_v q(G−v)=(n−2)(q−1) and Σ_v T(M−v)=(n−2)T(M)+d0 transfer mass from edges and diagonal a

What would settle it

Compute min{s+(G),s−(G)} exactly for every connected graph on up to n=10 vertices; any graph with value below n−1 would refute Theorem 1.2. Independently, solve the optimization over doubly nonnegative M: if for some connected graph the ratio 4(Σ_{edges}√M_uv)^2/(1^TM1) exceeds 2|E|−n+1, Theorem 2.1—the paper's main engine—is false, even if the spectral conjecture itself might survive.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: every connected graph G on n vertices satisfies min{s+(G), s−(G)} ≥ n−1, sharp for trees and for the negative square energy of K_n. The spectral statement is deduced from a standalone matrix inequality: for any connected graph G and any doubly nonnegative matrix M, 4(Σ_{uv∈E} √M_uv)^2 ≤ (2|E|−n+1)·1^TM1. This works because the Hadamard squares of the positive and negative spectral parts of the adjacency matrix are doubly nonnegative, and the sum of their edge entries dominates half the corresponding square energy. The matrix inequality is proved by induction on n: non-edge entries of M are folded onto the diagonal, cut vertices split the Gram representation

Load-bearing premise

The induction's no-cut-vertex case hinges on two exact deletion counts—each edge mass survives in all but two of the vertex-deleted submatrices and each diagonal entry in all but one—so a single miscount would break the averaged estimate and with it the proof of the main theorem.

Editorial extensions

If this is right

  • For connected G, s±(G) ≤ 2|E|−n+1, strengthening the classical spectral-radius bound λ1^2 ≤ 2|E|−n+1.
  • For a graph with κ components, min{s+, s−} ≥ n−κ, proving the disconnected version of the conjecture.
  • The squared spread satisfies |s+−s−| ≤ 2(|E|−n+κ); in particular it is at most (n−1)(n−2), with equality only for K_n.
  • The adjacency inertia bound min{s+,s−} ≥ n0 − ι + max{n+, n−} holds for every graph.
  • Positive and negative p-energies satisfy Eσ_p ≥ (n−κ)^{p/2} for 0<p≤2 and Eσ_p ≥ n−κ for p≥2, and the complement inequality sσ(G)+sσ(G̅) ≤ (n−1)^2 holds.

Reading between the lines

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

  • Editorial inference: because Theorem 2.1 is stated for all doubly nonnegative M, not just spectral Hadamard squares, it is a continuous inequality that can be tested or optimized by semidefinite programming independently of eigenvalues; any violation for a non-spectral M would signal a flaw in the proof's engine, even if the final theorem still held.
  • Editorial inference: the induction suggests a transferable scheme—if a quadratic graph functional satisfies a cut-vertex split and a controlled vertex-deletion average, it holds for all connected graphs; this may apply to other graph invariants such as Laplacian energies or degree-based functionals.
  • Editorial inference: the equality cases (trees, disjoint edges, complete graph) hint that the slack of the conjecture is governed by the cyclomatic number; one testable strengthening is min{s+,s−}−(n−κ) ≥ f(c(G)) for an explicit positive f.
  • Editorial inference: the same inequality might be applied to matrices that are positive semidefinite and nonnegative but unrelated to adjacency spectra, such as covariance or similarity matrices on a connected graph, yielding lower bounds on the sum over edges of their square roots.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper proves Conjecture 1.1 of Elphick, Farber, Goldberg, and Wocjan: every connected graph G on n vertices satisfies min{s+(G), s−(G)} ≥ n−1. The proof is built on a new inequality (Theorem 2.1) for doubly nonnegative matrices indexed by the vertices of a connected graph: 4(∑_{uv∈E(G)} √M_uv)^2 ≤ q(G) 1^T M 1, where q(G)=2|E|−|V|+1. The theorem is proved by induction with three steps: folding non-edge entries onto the diagonal, splitting at a cut vertex using a Gram-vector argument, and handling the no-cut-vertex case by averaging the induction hypothesis over vertex deletions. Section 3 applies the inequality to the Hadamard squares A+∘A+ and A−∘A− to obtain s±(G) ≤ q(G), which, together with s+(G)+s−(G)=2m, yields the conjectured lower bound. Section 4 derives consequences for disconnected graphs, squared spread, adjacency inertia, positive and negative p-energies, and graph complements. The paper also claims a Lean 4 formalization of the main theorem.

Significance. If the proof is correct, as my checking suggests, this settles a conjecture that has been open since 2016 and that has attracted substantial partial progress. The doubly nonnegative relaxation introduced in Theorem 2.1 is a genuinely new framework for square-energy estimates and is likely to have further applications; the companion paper on a Turán-type strengthening is evidence of its reach. The proof is self-contained, has no fitted parameters or ad hoc axioms, and the formal-verification claim, if reproducible, would make the result machine-checked. The consequences in Section 4, including previously conjectured bounds on p-energies and adjacency inertia, follow cleanly from the main theorem. This is a significant contribution to spectral graph theory.

minor comments (4)
  1. [§1.2, 'Bounding the optimum by duality'] The claim that minimizing Φ_G over correlation matrices is a semidefinite program is stated without details; the objective involves nonlinear terms 1/(1−C_uv). Since this subsection is explicitly motivational, please add a remark that the reformulation is only sketched, or give a reference for the SDP representability of the epigraph.
  2. [§1.3, 'Formal verification'] The manuscript states that Theorem 1.2 has been formally verified in Lean 4 but gives no commit hash or list of verified statements in the arXiv text. As a reproducibility matter, please include a specific version identifier and a brief description of what the formalization covers.
  3. [§2, no-cut-vertex case] The averaging step is compressed. In particular, the nonnegativity of q(G−v), which is needed for the Cauchy–Schwarz inequality over v, is not stated; it follows because G−v is connected on n−1 vertices. A one-sentence clarification would improve readability.
  4. [§4.4, proof of Corollary 4.4] For 0<p≤2, the inequality ∑ x_i^p ≥ (∑ x_i^2)^{p/2} is used without comment. It follows from ℓ_p/ℓ_2 norm monotonicity; adding a parenthetical would help readers not familiar with this fact.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 2.1 is proved by self-contained induction and Theorem 1.2 is a direct application to A±∘A±.

full rationale

The derivation chain is Theorem 2.1 -> Theorem 1.2. Theorem 2.1 is a standalone inequality for doubly nonnegative matrices, proved by induction on the number of vertices; the no-cut-vertex case uses explicit counting identities, including sum_v q(G-v) = (n-2)(q-1) and sum_v T(M-v) = (n-2)T(M)+d0, together with a Cauchy-Schwarz averaging argument, while the cut-vertex case uses a Gram-vector splitting. These arguments do not invoke Conjecture 1.1 or the square energies s±(G). Section 3 applies Theorem 2.1 to M = A+∘A+ and M = A-∘A-, using the Schur product theorem and the identities sum_edges |(A±)_uv| >= ±(1/2)<A,A±> = (1/2)s±(G); this is a direct reduction rather than a re-derivation of the input. No fitted parameter is renamed as a prediction. The self-citations that appear, e.g. [4], [9], [18], and [24], are contextual prior bounds, conjecture attributions, or companion-work references, and none is used as a premise of Theorem 2.1 or Theorem 1.2. The Lean formalization statement is a reproducibility claim, not part of the mathematical derivation. No circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof uses only standard matrix analysis and elementary graph counting; no parameters are fitted, and no new objects or entities are introduced.

assumptions (6)
  • standard math Schur product theorem: the Hadamard product of positive semidefinite matrices is positive semidefinite.
    Used in Section 3 to assert A±∘A± ⪰ 0.
  • standard math Spectral decomposition: A = A+ − A− with A+, A− ⪰ 0 and A+A− = 0.
    Defines s±(G) and is used in the edge-sum arguments of Section 3.
  • standard math Every symmetric positive semidefinite matrix has a Gram representation M_uv = ⟨z_u, z_v⟩.
    Used in the cut-vertex case of Theorem 2.1.
  • standard math Cauchy–Schwarz inequality.
    Used throughout the averaging steps and in the p-energy bounds.
  • domain assumption A connected graph with at least one edge has adjacency eigenvalues of both signs (trace zero and not identically zero).
    Used in Section 3 to ensure s+(G) > 0 and s−(G) > 0 before division.
  • standard math Hölder's inequality for p-energy bounds.
    Used in Corollary 4.4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The positive and negative square-energy conjecture." pith.science (2026). https://pith.science/paper/MJXZDAO7

@misc{pith2026260718031,
  author       = {Pith},
  title        = {Pith review of: The positive and negative square-energy conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJXZDAO7}},
  note         = {Machine review of arXiv:2607.18031}
}
abstract

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 7 linked inside Pith

  1. [1]

    Abiad, L

    A. Abiad, L. de Lima, D. N. Desai, K. Guo, L. Hogben, and J. Madrid, Positive and negative square energies of graphs,Electron. J. Linear Algebra39(2023), 307–326

    2023

  2. [2]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, and S. Pragada, A linear lower bound for the square energy of graphs, Electron. J. Combin.32(2025), no. 3, Paper No. P3.53

    2025

  3. [3]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, and S. Pragada, Vertex partitioning andp-energy of graphs,Linear Algebra Appl.724(2025), 96–107

    2025

  4. [4]

    Akbari, H

    S. Akbari, H. Kumar, B. Mohar, S. Pragada, and S. Zhang, Refinement of a conjecture on positive square energy of graphs, arXiv:2506.07264, 2025

    arXiv 2025

  5. [5]

    Ando and M

    T. Ando and M. Lin, Proof of a conjectured lower bound on the chromatic number of a graph,Linear Algebra Appl.485(2015), 480–484

    2015

  6. [6]

    Berman and N

    A. Berman and N. Shaked-Monderer,Completely Positive Matrices, World Scientific Publishing Co., River Edge, NJ, 2003

    2003

  7. [7]

    Z. Chen, Z. Wang, and X.-D. Zhang, Positive and negative3-energies of graphs, arXiv:2604.15656, 2026

    arXiv 2026

  8. [8]

    Coutinho and T

    G. Coutinho and T. J. Spier, Sums of squares of eigenvalues and the vector chromatic number, arXiv:2308.04475, 2023

    arXiv 2023

Show all 24 references
  1. [9]

    Coutinho, T

    G. Coutinho, T. J. Spier, and S. Zhang, Conic programming to understand sums of squares of eigenvalues of graphs, arXiv:2411.08184, 2024

    2024 arXiv

  2. [10]

    Elphick and M

    C. Elphick and M. Aouchiche, Nordhaus–Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph,Linear Algebra Appl.530(2017), 150–159

    2017

  3. [11]

    Elphick, M

    C. Elphick, M. Farber, F. Goldberg, and P. Wocjan, Conjectured bounds for the sum of squares of positive eigenvalues of a graph,Discrete Math.339(2016), no. 9, 2215–2223

    2016

  4. [12]

    Elphick and W

    C. Elphick and W. Linz, Symmetry and asymmetry between positive and negative square energies of graphs, Electron. J. Linear Algebra40(2024), 418–432

    2024

  5. [13]

    Elphick, Q

    C. Elphick, Q. Tang, and S. Zhang, A spectral lower bound on chromatic numbers usingp-energy,European J. Combin.132, Part B (2026), Article No. 104252

    2026

  6. [14]

    Hong, A bound on the spectral radius of graphs,Linear Algebra Appl.108(1988), 135–139

    Y. Hong, A bound on the spectral radius of graphs,Linear Algebra Appl.108(1988), 135–139

    1988

  7. [15]

    R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013

    2013

  8. [16]

    X. Li, Y. Shi, and I. Gutman,Graph Energy, Springer, New York, 2012

    2012

  9. [17]

    Liu and B

    L. Liu and B. Ning, Unsolved problems in spectral graph theory,Oper. Res. Trans.27(2023), no. 4, 33–60

    2023

  10. [18]

    Liu and Q

    Y. Liu and Q. Tang, Path-minimality for positivep-energies, Laplacian-type spectra, and line graphs, arXiv:2606.30996v1, 2026

    2026 arXiv

  11. [19]

    Y. Liu, Q. Tang, and S. Zhang, A positive square-energy strengthening of Turán’s theorem, preprint, 2026

    2026

  12. [20]

    Ning and J

    B. Ning and J. Zeng, A proof of a conjecture on positive and negative square energies of unicyclic graphs, arXiv:2605.24668, 2026

    2026 arXiv

  13. [21]

    Q. Tang, Y. Liu, and W. Wang, On the positive and negativep-energies of graphs under edge addition, Discrete Appl. Math.388(2026), 25–33. 12 Y. LIU, Q. TANG, AND S. ZHANG

    2026

  14. [22]

    Vandenberghe and S

    L. Vandenberghe and S. Boyd, Semidefinite programming,SIAM Rev.38(1996), no. 1, 49–95

    1996

  15. [23]

    Wocjan and C

    P. Wocjan and C. Elphick, New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix,Electron. J. Combin.20(2013), no. 3, Paper No. P39

    2013

  16. [24]

    Zhang, Extremal values for the square energies of graphs, arXiv:2409.15504v2, 2024

    S. Zhang, Extremal values for the square energies of graphs, arXiv:2409.15504v2, 2024. Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, P. R. China Email address:liuyinch23@mails.tsinghua.edu.cn School of Mathematical Sciences, Univers...

    2024 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.