{"id":"6bf9f9a7-1930-41e2-a102-a3f4bd6b7006","arxiv_id":"2506.07264","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For connected claw-free graphs with maximum degree at least 3 and for diameter-2 graphs other than stars and C5, the positive square energy is at least the number of vertices.","lead":"This mathematics paper proves that the positive square energy of a graph, the sum of squares of its positive eigenvalues, is at least the number of vertices for two large graph families: claw-free graphs and graphs of diameter 2. It introduces a gluing lemma that refines earlier super-additivity results and is expected to be reusable in other spectral graph problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unicyclic claw-free proof depends on unreleased computer checks and Desmos-verified inequalities, so the main theorems are not fully established until those numerical assertions are independently confirmed.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing issue: the main theorems rely on finite and numerical checks that are not reproducible from the manuscript. I focused on the most critical instance—Lemma 4.3 and Lemma 4.5 inside the proof of the unicyclic claw-free case—because Theorem 1.1 is the paper's headline result and the proof explicitly describes this case as the most challenging. The Desmos-verified inequalities are also load-bearing for Lemma 2.4, which feeds into Theorems 1.2 and 1.3. I did not find an internal contradiction in the gluing lemma or the case analysis; the argument structure appears coherent. The issue is verifiability of the computational assertions, not an identified error. A conditional acceptance with a request for code or analytic proofs is appropriate, so the reader's verdict remains unchanged.","tokens_in":21801,"tokens_out":20534,"duration_ms":198958,"concrete_test":"Write an independent script (e.g., Python with numpy or Sage) that computes the positive square energy of the explicit weighted matrices in Lemma 4.3 for every t in [2,600] and [10,600], verifies Lemma 4.4's small cases, computes s+(H(k,l)) for all k+l=n-5 with n<160, and checks the Desmos-used inequalities in Lemma 2.4, Lemma 4.2, and Lemma 4.5 via interval arithmetic or high-precision evaluation. If all checks pass, the numerical gap is closed; if any fail, the proof needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 rests on Theorem 4.1, whose proof depends on Lemma 4.3, a finite-range computer verification with no code or output provided, and on Lemma 4.5, whose two key inequalities are asserted only as 'checked using Desmos.' Lemma 4.4 additionally invokes an unshown computer check for ℓ=1 cases, and Proposition 7.1 relies on a computer check for n<160. Theorem 1.2(i) and Theorem 1.3(ii) further rely on Lemma 2.4, also justified by a Desmos-only inequality. If any of these checks is incorrect or incomplete, the corresponding theorem is not established as written. The paper gives no reproducible code, no exact arithmetic, and no analytic derivation for these assertions. While the checks are finite and probably correct, the written record does not let a reader confirm them, and the central claims of the paper depend on them.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22045,"tokens_out":24039,"duration_ms":237663,"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":[{"comment":"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.","section":"Abstract and Theorem 1.3"},{"comment":"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.","section":"Lemmas 4.2, 4.3, 4.5; Proposition 7.1; Theorem 4.1"},{"comment":"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.","section":"Theorem 6.1, Eq. (8)"}],"minor_comments":[{"comment":"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.","section":"Lemma 3.2"},{"comment":"There are several spacing and typesetting errors, such as 'ordern' and 'matrixA(G)', which should be corrected.","section":"Abstract and Introduction"},{"comment":"The sentence 'if there is a vertex u in V(G) with deg(v) >= 6' should read 'deg(u) >= 6'.","section":"Theorem 4.2, Case 2"},{"comment":"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}.","section":"Theorem 6.1"},{"comment":"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.","section":"Section 8"}],"recommendation":"major_revision","confidential_remarks":"The main risk is reproducibility: the paper relies on Desmos checks and finite computer verifications without code or outputs. Since the matrices involved have half-integer diagonal entries, exact-arithmetic verification is feasible and would considerably strengthen the manuscript. I would also ask the authors to correct the abstract's overclaim about Conjecture 1.1 for domination number at most 2. The citation pattern and the use of prior results appear appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main news: the paper proves the s+ >= n strengthening of the Elphick-Farber-Goldberg-Wocjan conjecture for claw-free graphs (with delta >= 3) and for diameter-2 graphs (except K1,n-1 and C5). The gluing lemma (Lemma 3.2) is the real contribution: it refines super-additivity by keeping track of diagonal corrections, and it looks independently useful. The diameter-2 proof in Section 6 is clean and elementary, and the domination-number results (Theorem 1.3) are solid extensions of Zhang's bound.\n\nWhere I would push back: the abstract claims to 'verify this conjecture' for domination number at most 2, but Theorem 1.3(ii) only proves the s+ side; the s- side for that family is still open. More importantly, the proof of Theorem 4.1, which is the heart of the claw-free result, rests on Lemma 4.3 (a computer check for t up to 600), Lemma 4.4 (more finite checks), Lemma 4.5 (two inequalities certified only by Desmos), and Proposition 7.1 (a check for n < 160). Lemma 2.4 also uses a Desmos-only verification. None of this is reproducible: no code, no exact arithmetic, no analytic proof. The checks are finite and probably correct, but they are load-bearing, and the paper as written doesn't let a reader confirm them. That is a real gap; I'm not saying the theorem is false, just that the proof isn't finished.\n\nI want to be fair: the rest of the argument is coherent. The gluing lemma's proof is valid, and the case analysis in Section 4 has the ring of a correct strategy. The references are appropriate; the overlap with [22] and [2] is contiguous work, not questionable fitting.\n\nBottom line: this is for spectral graph theorists. It deserves a serious referee, and I'd send it out. The referee should insist on reproducible verification for the numerical assertions, or analytic proofs for those inequalities, and a corrected abstract. If that's done, the strengthened conjecture for two major families is a strong result.","headline":"Strong result on square energy, but the flagship proof depends on Desmos checks and unreleased computer code.","tokens_in":22591,"tokens_out":6063,"would_cite":true,"duration_ms":55760,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C69","05C76"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["positive square energy","negative square energy","graph eigenvalues","claw-free graphs","diameter 2","domination number","super-additivity","P3-removal lemma"],"falsifier":"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.","tokens_in":21592,"feed_emoji":"🔢","tokens_out":13577,"duration_ms":108697,"temperature":0.7,"pith_summary":"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.","feed_headline":"Positive square energy at least n for two broad graph families","feed_subtitle":"Claw-free and diameter-2 graphs satisfy the strengthened bound s+(G) >= n, tightening a 2016 conjecture.","key_machinery":"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.","core_discovery":"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 $s^+(G) \\ge n$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"states the original conjecture $\\min\\{s^+, s^-\\} \\ge n-1$ that this paper refines and partially verifies.","marker":"[10]"},{"why":"supplies the super-additivity theorem, the original $P_3$-removal lemma, and the $s^+(G) \\ge n - \\gamma$ bound that the paper improves.","marker":"[22]"},{"why":"established the earlier linear lower bound via super-additivity, which the Gluing lemma strengthens.","marker":"[2]"},{"why":"provided known positive and negative square energy results, including cycles, that the paper extends and compares against.","marker":"[1]"},{"why":"interlacing theorem is used throughout to transfer positive eigenvalues from induced subgraphs to the host graph.","marker":"[15]"},{"why":"max-min formula for $\\lambda_1 + \\lambda_2$ is used in the domination-number-2 and $H(k,\\ell)$ arguments.","marker":"[9]"},{"why":"the rank of a tree equals twice its matching number, used to identify the two positive eigenvalues of a tree in the $H(k,\\ell)$ proof.","marker":"[7]"},{"why":"diameter-2 graphs have domination number $O(\\sqrt{n \\log n})$, used for the $s^-$ bound in Theorem 1.2(ii).","marker":"[8]"}],"fun_headline_variants":["Square energy bound strengthened for claw-free and diameter-2 graphs","Claw-free and diameter-2 graphs hit new square energy bound","Positive square energy conjecture refined for two graph classes","Proving s+(G) ≥ n for claw-free and diameter-2 graphs","Square energy lower bound lifted for two graph families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Square energy bound strengthened for claw-free and diameter-2 graphs","Claw-free and diameter-2 graphs hit new square energy bound","Positive square energy conjecture refined for two graph classes","Proving s+(G) ≥ n for claw-free and diameter-2 graphs","Square energy lower bound lifted for two graph families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00102,"raw_usage":{"total_tokens":4296,"prompt_tokens":927,"completion_tokens":3369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":3284}},"tokens_in":543,"tokens_out":3369,"duration_ms":22848,"temperature":1.0,"reasoning_tokens":3284,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:40:50.037524+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A Linear Lower Bound for the Square Energy of Graphs","cited_arxiv_id":"2409.18220","evidence_quote":"established the earlier linear lower bound via super-additivity, which the Gluing lemma strengthens."},{"cited_title":"Positive and negative square energies of graphs.Electron","cited_arxiv_id":null,"evidence_quote":"provided known positive and negative square energy results, including cycles, that the paper extends and compares against."},{"cited_title":"On the sum of two largest eigenvalues of a symmetric matrix","cited_arxiv_id":null,"evidence_quote":"max-min formula for $\\lambda_1 + \\lambda_2$ is used in the domination-number-2 and $H(k,\\ell)$ arguments."},{"cited_title":"Cvetković and Ivan M","cited_arxiv_id":null,"evidence_quote":"the rank of a tree equals twice its matching number, used to identify the two positive eigenvalues of a tree in the $H(k,\\ell)$ proof."},{"cited_title":"Graphs with diameter 2 and large total domination number.Graphs Combin., 37(1):271–279, 2021","cited_arxiv_id":null,"evidence_quote":"diameter-2 graphs have domination number $O(\\sqrt{n \\log n})$, used for the $s^-$ bound in Theorem 1.2(ii)."}],"review_version":1}