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Refinement of a conjecture on positive square energy of graphs

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Pith's one-line read The paper proves that every connected claw-free graph of order n with maximum degree at least 3, and every diameter-2 graph other than the star $K_{1,n-1}$ and the 5-cycle, has positive square energy at least n, confirming a strengthened…

desk verdict Strong result on square energy, but the flagship proof depends on Desmos checks and unreleased computer code. read the letter →

arxiv 2506.07264 v1 pith:5C2746OG submitted 2025-06-08 math.CO

Saieed Akbari , Hitesh Kumar , Bojan Mohar , Shivaramakrishna Pragada , Shengtong Zhang This is my paper · ORCID
classification math.CO MSC 05C5005C6905C76
keywords positivesquareenergynegativegrapheigenvaluesclaw-freegraphsdiameter2dominationnumbersuper-additivityP3-removallemma
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 tries to establish that the positive square energy of a graph, the sum of the squares of the positive eigenvalues of its adjacency matrix, is at least the number of vertices for broad families of connected graphs. This is a strict strengthening of an earlier conjecture that only claimed a lower bound of one less than the order. The authors prove it for all connected claw-free graphs with maximum degree at least 3 and for all diameter-2 graphs except the star and the 5-cycle, and they prove the weaker $n-1$ bound for every connected graph with domination number at most 2. The interest is that $s^+(G)$ packages spectral information about graph structure; a floor at the number of vertices would mean the positive spectrum alone carries enough energy to witness the graph's size.

What carries the argument

The load-bearing object is the Gluing lemma (Lemma 3.2), which refines super-additivity of square energy. If $G$ is formed by gluing graphs $G_1,\ldots,G_k$ onto a base graph $G_0$ at identified vertices, the lemma gives $s^+(G) \ge \sum_i s^+(G_i) + s^+(\Gamma)$, where $\Gamma$ is the adjacency matrix of $G_0$ with the diagonal entries at the gluing vertices replaced by the corresponding diagonal entries of $-A^-(G_i)$. This works through the identity $s^+(G)=\inf_{M \succeq 0} \|A(G)+M\|_F^2$, so positive square energy is a norm-minimization quantity. For the claw-free unicyclic case, applying the gluing lemma to a triangle with three attached paths reduces the theorem to checking the positive square energy of small weighted auxiliary graphs. A second tool, the improved $P_3$-removal lemma, says that deleting a suitable vertex from any induced $P_3$ loses at least $1+1/16$ of square energy; this carries the domination-number-2 and $\alpha\omega$ arguments.

What would settle it

Compute $s^+(P(j,k,\ell))$ exactly for a triple with $\min(j,k,\ell) > 600$, or verify the inequality $s^+(\Gamma_{t,3}) \ge 3(t+1) - 0.2$ at $t = 601$; if either fails, Theorem 4.1 and hence Theorem 1.1 collapse as written. More directly, a single connected claw-free graph with maximum degree at least 3 and $s^+(G) < n$, or a diameter-2 graph other than $K_{1,n-1}$ or $C_5$ with $s^+(G) < n$, would refute the strengthened conjecture.

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Extended reading notes

Core claim

For a simple graph $G$ with adjacency eigenvalues $\lambda_1 \ge \cdots \ge \lambda_n$, define $s^+(G)=\sum_{\lambda_i>0} \lambda_i^2$. The paper's central claim is that the strengthened inequality $s^+(G) \ge n$ holds for connected claw-free graphs with maximum degree at least 3 (Theorem 1.1) and for connected diameter-2 graphs other than $K_{1,n-1}$ and $C_5$ (Theorem 1.2(i)). It also proves $s^+(G) \ge n-1$ for every connected graph with domination number at most 2, and $s^+(G) \ge n$ for connected non-star graphs with a dominating vertex. The arguments introduce a Gluing lemma that refines the known super-additivity of square energy and an improved $P_3$-removal lemma, and the paper reads these as evidence for the stronger conjecture that every connected graph with at least $n+1$ edges has $s^+(G) \ge n$.

Load-bearing premise

The central claim relies on several numerical inequalities that are checked only for finite parameter ranges, with $t$ up to 600, $n$ under 160, and graphing software rather than analytic proofs; if any of those checks is wrong or incomplete, the main theorems are not established as written.

Editorial extensions

If this is right

  • Line graphs are claw-free, so every connected line graph of order $n$ with maximum degree at least 3 satisfies $s^+(G) \ge n$.
  • Because almost all graphs have diameter 2, the strengthened conjecture holds for almost all graphs, giving an independent route to a statement previously known through random-graph estimates.
  • The original conjecture $\min\{s^+(G), s^-(G)\} \ge n-1$ is verified, as the paper claims, for all connected graphs with domination number at most 2.
  • The improved $P_3$-removal lemma yields both $s^+$ and $s^-$ at least $n$ whenever $\alpha(G)\omega(G) \le n/17$, and $s^-(G) \ge n-1$ for graphs containing the 16th power of a Hamiltonian cycle.
  • For claw-free and diameter-2 families, the results settle the strengthened conjecture whenever the graph has at least $n+1$ edges, which is the regime the new conjecture targets.

Reading between the lines

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

  • The Gluing lemma is proved through a general norm-decomposition inequality, so the same mechanism should yield analogous lower bounds for $s^-$ and for other Schatten norms; testing it on cycles of length $4k+1$, where $s^+ < n$, would show exactly where the strengthened conjecture must stop.
  • The finite-range computer checks, with $t$ up to 600 and $n$ under 160, are the natural stress point: verifying Lemma 4.3 at $t=601$, or giving analytic proofs of the graphing-software inequalities, would convert the main theorems from computer-assisted to fully analytic.
  • If the strengthened conjecture holds for all graphs with $m \ge n+1$, then the original $n-1$ bound follows automatically for every connected graph with a cycle, and the paper's proposed equality cases, bipartite unicyclic graphs, would describe the exact boundary between the $n$ and $n-1$ regimes.
  • The diameter-2 proof is quantitative, so it suggests a search for diameter-2 graphs with very few triangles and induced 4-cycles; those should have $\lambda_1^2$ just above $n$ and could reveal how close the bound is to being tight.
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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 / 5 minor

Summary. This paper studies the sums of squares of positive and negative eigenvalues of a graph, s+(G)=sum over positive eigenvalues of lambda_i^2 and s-(G)=sum over negative eigenvalues of lambda_i^2. It proposes a strengthening (Conjecture 1.2) of the Elphick-Farber-Goldberg-Wocjan conjecture: every connected graph with m>=n+1 satisfies s+(G)>=n. The main results are: Theorem 1.1, Conjecture 1.2 for claw-free graphs with maximum degree at least 3; Theorem 1.2, Conjecture 1.2 for graphs of diameter 2 and a lower bound s-(G)>=n-O(sqrt(n) log n) for such graphs; Theorem 1.3, s+(G)>=n-1 for graphs with domination number at most 2 and s+(G)>=n for non-star graphs with a dominating vertex; and Theorem 8.1, min{s+,s-}>=n when alpha(G)omega(G)<=cn. The main technical novelties are a strengthened P3-removal lemma and a gluing lemma refining super-additivity of square energy. The proof of the unicyclic claw-free case relies on several computer and Desmos-based checks.

Significance. The results, if correct, are significant: claw-free graphs and diameter-2 graphs are broad families, and Conjecture 1.2 has not previously been known for them. The gluing lemma is a genuinely new tool that refines super-additivity and is likely to have further applications; the paper also offers clean open conjectures and exact computations for cycles. The structural arguments are mostly clear, and the cited prior work is used appropriately. However, several load-bearing numerical assertions are not reproducible from the manuscript, and one abstract claim is stronger than what is proved. These issues should be resolved before the paper can be accepted.

major comments (3)
  1. [Abstract and Theorem 1.3] The abstract states that Conjecture 1.1 is verified for graphs with domination number at most 2, but Theorem 1.3(ii) only proves s+(G)>=n-1 for gamma(G)<=2. Conjecture 1.1 requires min{s+(G),s-(G)}>=n-1. No s- lower bound of n-1 for gamma=2 is proved: Theorem 5.1 gives n-2, and the proof of Theorem 7.1 is not symmetric, since the s- analogue of Theorem 5.2 is false (K4 has gamma=1 and s-(K4)=3<4). The authors should either supply an s- proof for gamma=2 or change the abstract and introduction to claim only the positive part of Conjecture 1.1.
  2. [Lemmas 4.2, 4.3, 4.5; Proposition 7.1; Theorem 4.1] The proofs of Theorems 1.1 and 1.3(ii) depend on numerical assertions that are not verifiable from the manuscript. Lemma 4.2 is justified by analyzing a function in Desmos; Lemma 4.5 uses Desmos for inequalities (2) and (4); Lemma 4.3 states computer verification for t in [2,600] and t in [10,600] with no code or output; Lemma 4.4 asserts the ell=1 cases via the matrices Gamma_a and Gamma_b and declares the case j=k=2 checkable explicitly without displaying the values; Proposition 7.1 uses a computer check for n<160 and the value lambda_3(H(1,1))>=0.71 without derivation. Since Theorem 4.1 is the core of Theorem 1.1 and Proposition 7.1 feeds into Theorem 7.1, the central claims are not established as written. Please provide exact-arithmetic computer code and outputs, or replace each Desmos or computer check by an analytic proof.
  3. [Theorem 6.1, Eq. (8)] The passage from (6) to (8) is not proved. The claims that each induced C4 allows one to add 2 units to the left side of (6) and each triangle allows one to reduce the right side by 3 units are stated as 'not hard to see' but are load-bearing for Theorem 1.2(i), since the case analysis repeatedly invokes (8). A short rigorous derivation should be included. In addition, the proof begins 'We can assume n>=4' without treating the n=3 case (K3 has diameter 2 and is not excluded); this should be closed explicitly.
minor comments (5)
  1. [Lemma 3.2] The displayed definition of R1 uses (A-(G))_{u,u'} while the proof applies Lemma 3.1 to G_i and then uses (A-(G_i))_{u,u'}; the notation should be made consistent throughout the statement and proof.
  2. [Abstract and Introduction] There are several spacing and typesetting errors, such as 'ordern' and 'matrixA(G)', which should be corrected.
  3. [Theorem 4.2, Case 2] The sentence 'if there is a vertex u in V(G) with deg(v) >= 6' should read 'deg(u) >= 6'.
  4. [Theorem 6.1] The phrase 'We can assume n>=4' should be accompanied by an explicit treatment of n=3, since K3 is a connected graph of diameter 2 not in {K_{1,n-1}, C5}.
  5. [Section 8] The assertion that each of the listed r-vertex graphs H has s+(H)>=4r/3 is stated without computation; a one-line verification for each graph would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proved by new derivations, and the cited author-prior results are parameter-free lemmas that do not contain the target inequalities.

full rationale

The paper's central results are not obtained by fitting a parameter and then predicting the same quantity. Theorem 1.1 uses the new Gluing Lemma 3.2, which is proved in the paper, together with analytic and computer-assisted estimates; Theorem 1.2 is derived from elementary counting and interlacing; Theorem 1.3 and Theorem 7.1 use induction plus the strictly weaker prior bound s^+(G) >= n - gamma(G) from [22]. The self-citations to [2,22] for super-additivity (Theorem 3.1), the PSD observation (Lemma 3.1), and the P3-removal lemma are to parameter-free statements whose assumptions do not include the target bounds, so under the stated rules they count as independent support and do not raise the circularity score. The Desmos checks in Lemmas 2.4, 4.2, and 4.5, and the finite computer checks in Lemmas 4.3, 4.4, and Proposition 7.1, are reproducibility or verification concerns: they are explicit numerical assertions about fixed analytic quantities, not circularly defined predictions, and no equation in the paper reduces to its input by construction. The skeptical reading is about unverified computation, not about circularity, so the appropriate circularity score is 0.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The proofs are largely analytic, but the central unicyclic case and the improved P3-removal lemma depend on finite computer checks and Desmos-verified inequalities. The chosen constants and thresholds are listed as free parameters because they are load-bearing for the proof, and the Desmos dependence is listed as an ad hoc axiom.

free parameters (5)
  • epsilon in Lemma 2.4 (improved P3-removal) = 1/16
    Chosen so that the polynomial inequality 16x^4 > 6(1+epsilon-4(1-x)^2)(1+epsilon-2(1-x)^2) holds on the stated interval; the paper validates this only with Desmos.
  • constant c in Theorem 8.1 = 1/17
    Derived as epsilon/(1+epsilon) from epsilon = 1/16; not fitted to data, but it is an input value chosen in the proof.
  • diagonal-entry bounds 0.5 and 0.43 in Lemma 4.2 = 0.5, 0.43
    Upper bounds on (A-(P_l)) at a leaf; the paper states these are verified by analyzing functions in Desmos rather than by a written proof.
  • off-diagonal lower bound 0.21 in Lemma 4.5 = 0.21
    Lower bound on (A-(P_l))_{j,j+2} for central j; the main term is checked via Desmos, with an analytic error estimate.
  • thresholds 0.44 and 0.57 in Theorem 4.1 = 0.44, 0.57
    Chosen constants used in the R1/R2 Cauchy-Schwarz estimates and in the diagonal entries of the auxiliary matrices; the supporting finite computations in Lemma 4.3 are described but not released.
assumptions (7)
  • standard math Interlacing theorem for principal submatrices
    Theorem 2.1, used throughout the proof via interlacing of eigenvalues.
  • standard math Weak majorization and Lp norm inequality
    Theorem 2.2 is used in Lemma 7.1 and Proposition 7.1 to convert weak majorization into a sum-of-squares inequality.
  • standard math Max-min characterization of lambda1+lambda2
    Lemma 2.3, cited from [9], is used to prove lower bounds on lambda1+lambda2 in Sections 5 and 7.
  • standard math Tree rank equals twice the matching number
    Theorem 2.3 is used in Proposition 7.1 to identify that the auxiliary tree T has exactly two positive eigenvalues.
  • standard math Spectrum of joins of regular graphs
    Theorem 2.4 is used in the open problems section for examples on maximal planar graphs.
  • domain assumption Super-additivity of square energy
    Theorem 3.1 from [2,22] is used as a black box; it is a proven derivation, and the overlap in authorship does not make it circular.
  • ad hoc to paper Desmos graphing calculator is a reliable oracle for the inequalities in Lemmas 2.4, 4.2, and 4.5
    The paper repeatedly relies on statements such as 'A direct computation via Desmos shows...' or 'analyze the behaviour in Desmos' to establish key inequalities; no analytic proof or reproducible script is supplied.

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Pith. "Pith review of Refinement of a conjecture on positive square energy of graphs." pith.science (2026). https://pith.science/paper/5C2746OG

@misc{pith2026250607264,
  author       = {Pith},
  title        = {Pith review of: Refinement of a conjecture on positive square energy of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5C2746OG}},
  note         = {Machine review of arXiv:2506.07264}
}
abstract

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \geq n+1$, then $s^+(G) \geq n$. We prove this conjecture for claw-free graphs and graphs with diameter 2.

Figures

Figures reproduced from arXiv: 2506.07264 by the authors.

Figure 1
Figure 1. The graph G0 We now deal with the general case P(j, k, ℓ). First, we present some numerical results which are verified using a computer. Lemma 4.3. Let t be a positive integer. The following are true. (i) Consider the graph Gt,2 as shown in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 4
Figure 4. Graph H 5 Graphs with domination number 1 It is easy to see that if γ(G) = 1 for some graph G of order n, then λ 2 1 (G) ≥ n − 1. Using this fact and the super-additivity result (Theorem 3.1), Zhang [22] proved the following. Theorem 5.1 ([22]). Let G be a graph on n vertices and domination number γ(G). Then s +(G) ≥ n − γ(G) and s −(G) ≥ n − γ(G). In this section, we improve Theorem 5.1 for s + of connected graphs … view at source ↗

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Forward citations

Cited by 2 Pith papers

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

  1. A positive square-energy strengthening of Tur\'an's theorem

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    Every n-vertex graph with clique number ω has √s⁺(G) ≤ (1−1/ω)n, where s⁺(G) is the sum of squared positive adjacency eigenvalues.

  2. The positive and negative square-energy conjecture

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    Every connected graph G on n vertices satisfies min{s+(G), s−(G)} ≥ n−1, confirming the Elphick–Farber–Goldberg–Wocjan conjecture.

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