{"id":"3882c565-a009-4d77-8949-1303c075d523","arxiv_id":"2412.13792","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum spectral radius of an F6-free graph with m≥88 edges is (1+√(4m−3))/2, attained only by K2 ∨ ((m−1)/2)K1.","lead":"This paper proves the last open case of a spectral Turán conjecture about graphs that avoid a six-vertex fan shape. It gives the exact maximum spectral radius and the unique graph that achieves it for every graph with at least 88 edges.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 is not well-defined for even m: the extremal graph K2∨((m−1)/2)K1 does not exist for such m, so the equality statement and the proof's initial lower bound fail for half the claimed range.","rationale":"The reader's weakest-assumption identification is Lemma 3.2, the unproved classification of G∗[U]-components as star, double star, K1,r+e, C4, K4−e, or K4. This is indeed load-bearing: every later elimination lemma assumes exactly that list. I agree that the paper should either prove Lemma 3.2 or verify that Lemma 4.5 of [8] applies verbatim. However, I regard the parity defect as more fundamental, because it affects the theorem statement itself rather than only the completeness of a borrowed proof. For even m, the graph K2∨((m−1)/2)K1 is undefined, so the claimed equality is not a statement about graphs, and the proof's opening lower bound cannot be justified. The proof's algebra appears internally consistent modulo the borrowed classification; I did not find a clear fatal error in the Perron-vector estimates, although one minor false overstatement appears in Lemma 3.1 when it says λ > √m + 3 (the needed consequence m+3 < λ² is nevertheless true). The false overstatement is not load-bearing. A corrected theorem restricted to odd m, together with a proof of Lemma 3.2, would make the central claim sound; for even m a separate strict-inequality argument would be required. Since these are fixable conditions rather than a demonstration that the main bound is false, the appropriate verdict remains conditional.","tokens_in":19221,"tokens_out":34819,"duration_ms":302728,"concrete_test":"Set m = 88 (even) and check whether the proof's initial inequality λ(G∗) ≥ (1+√349)/2 has any valid justification: the graph K2∨(87/2 K1) is undefined. Then determine whether the intended domain is odd m; if so, restate the theorem as m = 2s+1, verify the proof for m = 89, and supply a separate proof for even m, e.g., by deleting one edge from an even-m F6-free graph and applying the odd-m case to the resulting (m−1)-edge graph, since edge deletion cannot increase the spectral radius. This distinguishes a statement-level typo from a genuine gap in the extremal argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem as stated applies to every m ≥ 88, but the extremal graph K2∨((m−1)/2)K1 has integral order only when m is odd. For even m, say m = 88, the graph is undefined, so the equality condition is not a well-formed graph isomorphism. This is not merely cosmetic. The proof opens by asserting λ(G∗) ≥ λ(K2∨((m−1)/2)K1) = (1+√(4m−3))/2, and it uses this lower bound to derive the key inequalities λ > 49/5 and λ² − λ ≥ m − 1 that drive every estimate from Lemma 3.1 onward. For even m no such graph exists, so the lower bound cannot be invoked and the extremal analysis has no valid starting point. Moreover, the final step concludes G∗ ≅ K2∨((m−1)/2)K1 and m = 2r+1, so the classification can only produce odd m. Thus Theorem 1.2 either needs an explicit hypothesis that m is odd (m = 2s+1, s ≥ 44), or it needs a separate argument for even m showing that λ is strictly below the stated bound. As written, the theorem is ill-formed for even m and the proof does not cover them.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the spectral radius of F6-free graphs with a fixed number of edges m, where F6 is the fan graph K1 ∨ P5. The main result, Theorem 1.2, claims that for every m ≥ 88, every F6-free graph G with m edges satisfies λ(G) ≤ (1 + √(4m−3))/2, with equality if and only if G ≅ K2 ∨ ((m−1)/2)K1. The proof takes an extremal graph G*, uses the Perron vector to derive a key inequality (1), and then analyzes the induced subgraph G*[U] on the neighborhood of a maximal Perron-coordinate vertex u*. After proving e(U) ≥ 4, the authors classify the components of G*[U] into six types, show γ(H) ≤ 0 for every nontrivial component, and then successively eliminate K4, K4−e, C4, and K1,r+e components, leaving only stars and double stars. A final argument forces the unique nontrivial component to be a star and W to be empty, yielding the extremal graph K2 ∨ ((m−1)/2)K1. The paper thus claims to resolve the remaining k = 2 case of a conjecture of Yu et al. on fan-free graphs.","tokens_in":19532,"tokens_out":5815,"duration_ms":51583,"significance":"If Theorem 1.2 is correct, the paper closes the last open case of a natural spectral Turán-type conjecture for fan graphs, complementing the recently proved cases k ≥ 3 by Li et al. The proof is a substantial, mostly self-contained case analysis built on the Perron-vector method, and the numerical thresholds (λ > 49/5, e(W) ≤ 1, etc.) are derived from the stated inequalities rather than fitted to the conclusion. The paper also gives an explicit extremal construction and identifies the unique extremal graph. The main significance is therefore conditional on two points that need attention: the well-definedness of the extremal graph and the validity of the component classification. The proof does not appear to be circular; the central bound is derived from the Perron-vector equations, and the cited references are used for supporting structural facts, not for the main F6 bound itself.","major_comments":[{"comment":"The theorem as stated is not well-defined for even m. For even m, (m−1)/2 is not an integer, so the graph K2 ∨ ((m−1)/2)K1 does not exist and the equality condition is not a well-formed graph isomorphism. The proof also uses this graph at the outset to assert λ(G*) ≥ (1+√(4m−3))/2, which yields the crucial inequalities λ²−λ ≥ m−1 and λ > 49/5. For even m this lower bound cannot be invoked. Moreover, the final step of the proof concludes m = 2r+1, so the argument as written only covers odd m. The theorem, the abstract, and the proof need either an explicit hypothesis that m is odd, or a separate treatment of even m showing that the upper bound still holds (and that equality is impossible). This is a load-bearing issue because every later estimate in the proof relies on the initial lower bound.","section":"Theorem 1.2"},{"comment":"Lemma 3.2 classifies every component of G*[N(u)] into six types (star, double star, K1,r+e, C4, K4−e, K4), but the proof is not given; the text says only 'Similarly to the proof of Lemma 4.5 in [8]' without stating the referenced lemma or explaining how it applies to P5-free induced neighborhoods. This classification is load-bearing: Lemmas 3.8, 3.9, 3.11, and 3.14 eliminate components by enumerating exactly these six types, and the final structural conclusion depends on the classification being complete. If the classification is incomplete or the hypotheses of the referenced result are not met, the proof does not cover all F6-free graphs. The authors should either provide a self-contained proof of Lemma 3.2 or give a precise statement of the relevant result from [8] and verify its applicability here.","section":"Lemma 3.2"}],"minor_comments":[{"comment":"The displayed inequality 'λ > √m + 3' appears to be a typesetting error; from λ ≥ (1+√(4m−3))/2 one can derive λ > √(m+3) for m ≥ 88, but not λ > √m + 3 (which is false for m = 88). Please correct the square-root notation in this line.","section":"Lemma 3.1"},{"comment":"The conjecture as quoted has the same integrality issue: the expression m/k − (k−1)/2 is not an integer for general m, so the extremal graph K_k ∨ (m/k − (k−1)/2)K1 is not always defined. Clarify the intended arithmetic condition (e.g., m ≡ k(k−1)/2 mod k) in the conjecture and in the abstract.","section":"Introduction, Conjecture 1.1"},{"comment":"The equality condition of inequality (2) states that equality holds if and only if λ²−λ = m−1 and xw = xu* for every w ∈ W with dU(w) ≥ 1; this condition is used later in Lemma 3.10 and in the final star analysis, so it would help to display it as a numbered equation for easier reference.","section":"Section 3"}],"recommendation":"major_revision","confidential_remarks":"The parity issue in Theorem 1.2 is not merely cosmetic: it affects the validity of the initial lower bound and hence the proof for even m. The authors should be asked to state the theorem only for odd m or to extend the argument. The reliance on Lemma 3.2 from reference [8] should also be made self-contained or precisely cited. If these two points are resolved, the main argument appears interesting and likely correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Gao and Li close the F6 gap in Yu et al.'s spectral Turán conjecture, and for odd m the proof looks sound. The Perron-vector framework is applied carefully: inequality (1) is derived from the eigenvector equations, the threshold λ>49/5 controls the arithmetic, and the elimination of K4, K4−e, C4, K1,r+e components forces the extremal graph to be K2∨((m−1)/2)K1. I checked the main estimates in Lemmas 3.1–3.4 and the final star/double-star step; nothing circular jumps out, and the bound is not fitted to the target.\n\nThe soft spots are real but fixable. First, Theorem 1.2 states m≥88 with no parity condition, but the extremal graph K2∨((m−1)/2)K1 exists only for odd m. The proof opens by using this graph to get the lower bound λ(G*)≥(1+√(4m−3))/2, so for even m the argument has no starting object and the claimed theorem is not proved. The last line of the proof even derives m=2r+1, so the argument only covers odd m. The statement should either restrict to odd m (m=2s+1, s≥44) or give a separate argument for even m.\n\nSecond, Lemma 3.2, which classifies every component of G*[U] as a star, double star, K1,r+e, C4, K4−e, or K4, is borrowed without proof ('Similarly to Lemma 4.5 in [8]'). Every later elimination depends on this list being exhaustive. It is probably true, and a precise citation to the exact statement in [8] might be enough, but as written the provenance is too vague for a lemma that carries the proof.\n\nMinor: the equality statement in the abstract repeats the same half-integer issue, and the notation (m−1)/2 K1 should be made unambiguous.\n\nOverall: the odd-m result is a genuine completion of the last open case, and the proof is a serious piece of work. With the parity hypothesis fixed and Lemma 3.2 properly sourced, I would take it. For peer review, yes—send it to a referee; the referee should focus on the parity restriction and the classification lemma.","headline":"The F6 case is genuinely solved for odd m, but the theorem as stated overclaims even m and the component classification needs a real proof or citation.","tokens_in":20028,"tokens_out":9319,"would_cite":true,"duration_ms":80263,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $m \\geq 88$, every $F_6$-free graph has spectral radius at most $(1+\\sqrt{4m-3})/2$, attained exactly by the book graph $K_2 \\vee ((m-1)/2)K_1$.","keywords":["spectral radius","fan graph","F6-free graph","extremal graph","Perron vector","P5-free graph","Brualdi–Hoffman–Turán problem"],"falsifier":"A concrete way to test the theorem is computational: for each $m$ from 88 up to, say, 200, enumerate or sample $F_6$-free graphs with $m$ edges, compute their spectral radii, and check that none exceeds $(1+\\sqrt{4m-3})/2$ and that the only graphs attaining it are isomorphic to $K_2 \\vee ((m-1)/2)K_1$. An alternative structural check is to exhibit an $F_6$-free graph whose neighborhood subgraph at some vertex has a connected $P_5$-free component that is not a star, double star, $K_{1,r}+e$, $C_4$, $K_4-e$, or $K_4$, which would falsify Lemma 3.2.","tokens_in":19023,"feed_emoji":"🔺","tokens_out":15279,"duration_ms":111195,"temperature":0.7,"pith_summary":"The paper settles the last open case of a spectral Turán conjecture for fan graphs, proving an exact upper bound on the spectral radius of any graph with $m \\geq 88$ edges that contains no copy of $F_6$, the fan graph formed by joining a new vertex to a path on five vertices. The bound is $\\lambda(G) \\leq (1+\\sqrt{4m-3})/2$, and equality holds exactly when $G$ is a book graph: two adjacent universal vertices joined to $(m-1)/2$ isolated vertices. This matters because it completes the conjecture for all fan graphs $F_k$, showing that the same family of complete split graphs is extremal in the one remaining case, $k=2$. The proof is an extremal-graph analysis using the Perron vector of the adjacency matrix.","feed_headline":"For m ≥ 88, F6-free graphs peak at the book graph","feed_subtitle":"The bound is sharp and settles the last open fan-graph spectral case.","key_machinery":"The proof's load-bearing device is a partition of the extremal graph $G^*$ at a vertex $u^*$ of maximal Perron coordinate: $U$ is the neighborhood of $u^*$, $W$ is the remaining vertices, and Perron-vector inequalities yield a key counting inequality that bounds the number of edges inside $W$. A second critical ingredient is Lemma 3.2, which classifies every connected component of $G^*[U]$ — necessarily $P_5$-free — as exactly one of six types: a star, a double star, a star with one added edge ($K_{1,r}+e$), $C_4$, $K_4-e$, or $K_4$. The subsequent lemmas eliminate all component types except stars, forcing the whole graph into the split form $K_2 \\vee ((m-1)/2)K_1$. The Perron-vector edge-shift lemma, which transfers edges toward higher-coordinate vertices and strictly increases the spectral radius, drives the elimination arguments.","core_discovery":"The central claim is Theorem 1.2: if $G$ is $F_6$-free and has $m \\geq 88$ edges, then its spectral radius $\\lambda(G)$ is at most $(1+\\sqrt{4m-3})/2$, and this value is attained exactly by the graph $K_2 \\vee ((m-1)/2)K_1$, a clique of two vertices whose common neighborhood is an independent set of size $(m-1)/2$. This confirms the $k=2$ case of the earlier conjecture that, for sufficiently many edges, $F_{2k+1}$-free and $F_{2k+2}$-free graphs have spectral radius bounded by $(k-1+\\sqrt{4m-k^2+1})/2$, with equality only on $K_k$ joined to an independent set.","pith_inferences":["If Lemma 3.2's six-type classification is made fully self-contained, the rest of the proof requires only the Perron-vector shift lemma and elementary counting, so the argument could be re-exposed as a standalone proof for $F_6$.","The same partition-and-classify strategy could in principle be adapted to the remaining small fan graphs $F_7$ or $F_8$, provided the corresponding neighborhood subgraph classification is worked out.","Computational searches for $m$ between roughly 20 and 87 could show whether the book graph remains extremal below the paper's threshold of 88, which the proof's constants suggest is not optimal."],"forward_implications":["The spectral Turán conjecture for fan graphs is now settled for every $k \\geq 2$, closing the $k=2$ gap left by the earlier unified proof for $k \\geq 3$.","Any $F_6$-free graph with $m \\geq 88$ edges and spectral radius greater than $(1+\\sqrt{4m-3})/2$ must contain a fan $F_6$ as a subgraph, so the bound is a sharp spectral-forcing threshold.","The extremal graph is unique up to isomorphism: $K_2 \\vee ((m-1)/2)K_1$, a clique of two vertices joined to an independent set of $(m-1)/2$ vertices.","For $m \\geq 88$, the bound implies $\\lambda(G) \\leq \\sqrt{m} + 1/2$, an asymptotic form useful for quick estimates."],"supporting_citations":[{"why":"Classifies connected P5-free components into the six types used by Lemma 3.2; the whole component-elimination chain depends on this classification.","marker":"[8]"},{"why":"Provides Lemma 2.4, that G[N(u)] is P_{k−1}-free in F_k-free graphs, used to show G*[U] is P5-free.","marker":"[5]"},{"why":"Supplies the Perron-vector edge-shift lemma used in every maximality contradiction.","marker":"[18]"},{"why":"Gives the structural lemma that an extremal graph is connected and that vertices outside the closed neighborhood of the extremal vertex have degree at least 2.","marker":"[20]"},{"why":"One of the two sources for the K3-free spectral bound λ(G) ≤ √m used to dismiss the bipartite case.","marker":"[11]"},{"why":"The other source for the same K3-free bound λ(G) ≤ √m.","marker":"[15]"}],"fun_headline_variants":["Max spectral radius for F6-free graphs: found for m≥88","Fan graph F6 avoided? Then λ ≤ (1+√(4m−3))/2","Last open fan case cracked: extremal graph is K2 plus independent set","F6-free graphs: sharp spectral bound for large m","Spectral radius peak for F6-free graphs: the book graph"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on Lemma 3.2, which states that every component of the neighborhood-induced subgraph $G^*[U]$ must belong to one of six listed graph families; this classification is borrowed from another paper with only a 'similarly' argument, and every subsequent elimination step enumerates exactly those six possibilities, so if the classification misses a possible component, the proof does not cover all $F_6$-free graphs.","fun_headline_variants_meta":{"raw":{"variants":["Max spectral radius for F6-free graphs: found for m≥88","Fan graph F6 avoided? Then λ ≤ (1+√(4m−3))/2","Last open fan case cracked: extremal graph is K2 plus independent set","F6-free graphs: sharp spectral bound for large m","Spectral radius peak for F6-free graphs: the book graph"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1434,"prompt_tokens":1027,"completion_tokens":407,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":322}},"tokens_in":643,"tokens_out":407,"duration_ms":4385,"temperature":1.0,"reasoning_tokens":322,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:47:26.174588+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the theorem is computational: for each $m$ from 88 up to, say, 200, enumerate or sample $F_6$-free graphs with $m$ edges, compute their spectral radii, and check that none exceeds $(1+\\sqrt{4m-3})/2$ and that the only graphs attaining it are isomorphic to $K_2 \\vee ((m-1)/2)K_1$. An alternative structural check is to exhibit an $F_6$-free graph whose neighborhood subgraph at some vertex has a connected $P_5$-free component that is not a star, double star, $K_{1,r}+e$, $C_4$, $K_4-e$, or $K_4$, which would falsify Lemma 3.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies connected P5-free components into the six types used by Lemma 3.2; the whole component-elimination chain depends on this classification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Perron-vector edge-shift lemma used in every maximality contradiction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the structural lemma that an extremal graph is connected and that vertices outside the closed neighborhood of the extremal vertex have degree at least 2."},{"cited_title":"Nikiforov, Some inequalities for the largest eigenvalue of a gra ph, Combin","cited_arxiv_id":null,"evidence_quote":"One of the two sources for the K3-free spectral bound λ(G) ≤ √m used to dismiss the bipartite case."}],"review_version":1}