{"id":"bc962d2b-914b-4e2d-aec7-dd9b65341e12","arxiv_id":"1908.04221","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For large orders, the unique extremal graphs for the signless Laplacian spectral radius among K2,t-minor free and K3,3-minor free graphs are F2,t(n) and F3,3(n), respectively.","lead":"This paper determines which graphs that avoid K2,t or K3,3 as a minor have the largest signless Laplacian eigenvalue, for large orders. The maximizing graph is always a specific join of a small clique with disjoint triangles, plus a remainder.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 rests on Lemmas 2.5–2.8, which are asserted to follow from [16] with no proof or exact reference; if any adaptation fails, the K3,3 extremal result collapses.","rationale":"I read the paper as a solid spectral extremal graph theory contribution whose main gap is the unproved adaptation of Lemmas 2.5–2.8 from [16]. The K2,t results (Theorems 1.2 and 1.3) are essentially self-contained modulo standard cited results, and the degree-sequence and edge-swapping arguments for them appear coherent. For Theorem 1.4, however, the proof relies structurally on the four borrowed lemmas to rule out all cases except two universal vertices. Since the source paper is about planar graphs and the present setting allows denser K3,3-minor-free graphs, the modified constants are not automatic; the authors' assertion that the lemmas 'are correct according to original proofs' is not a substitute for a proof or an exact citation. This is the same weakest assumption the reader identified, and I agree with the conditional verdict. I do not see an internal contradiction or a demonstrably false step, so I would not move the verdict to reject; I would keep it conditional pending a supplied proof of the adapted lemmas. A targeted analytical reconstruction of the original proofs is the cleanest way to settle the question, with a numerical search over degree sequences as a useful secondary check.","tokens_in":11640,"tokens_out":41055,"duration_ms":426224,"concrete_test":"Independently reconstruct the proofs of Lemmas 2.5–2.8 from the original paper [16], tracking every occurrence of the planar edge bound e(G) ≤ 3n−6. For each lemma, either transcribe the proof if it uses only the stated degree-sequence hypotheses, or replace the planar edge bound by e(G) ≤ 3n−5 and re-verify the inequalities that yield the constants n ≥ 115, n ≥ 91, k ≤ 12, and k ≤ 13. If any step uses planarity in a way that cannot be replaced by the weaker edge bound, Lemmas 2.5–2.8 are not established for K3,3-minor-free graphs and Theorem 1.4 remains conditional; if all steps go through, the adaptation is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is exactly the one the reader identifies: Lemmas 2.5–2.8 are imported from [16] in forms the authors call 'a little different from their original forms,' and the paper does not prove them or point to a theorem number where the modified statements appear. These four lemmas are essential: Lemma 4.2 Case 1 uses Lemma 2.5 and Lemma 2.6, and Lemma 4.3 Case 1 uses Lemma 2.7 and Lemma 2.8, to conclude q(G) ≤ n+2 for edge-maximal K3,3-minor-free graphs under the relevant degree-sequence cases. Without q(G) ≤ n+2 in those cases, the proof of Theorem 1.4 cannot force Δ(G)=Δ'(G)=n−1, and the whole reduction to two universal vertices and the extremal graph F3,3(n) fails. The missing part is nontrivial because the source [16] concerns planar graphs, whose edge bound is e(G) ≤ 3n−6, while K3,3-minor-free graphs can have e(G) ≤ 3n−5; the adapted constants in Lemmas 2.5 and 2.7 (the ranges n/6+1 to n−61, n/7+19/7 to n−75, and the bounds k ≤ 12, k ≤ 13) are precisely what make the edge-count inequalities in Lemmas 4.2 and 4.3 work. The paper gives no independent verification, no machine-checked proof, and no reproducible code, so the correctness of the adaptation is a genuine unverified step. Because all other parts of the argument appear internally coherent, this is a proof gap rather than a demonstrated contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the signless Laplacian spectral radius q(G) of K_{s,t}-minor-free graphs. Its main results are: (i) for K_{2,3}-minor-free graphs of order n≥22, q(G) ≤ q(F_{2,3}(n)) with equality if and only if G=F_{2,3}(n); (ii) for t≥4 and K_{2,t}-minor-free graphs of order n≥t^2+4t+1, q(G) ≤ (n+2t−2+√((n−2t+2)^2+8t−8))/2 with equality if and only if n≡1 (mod t) and G=F_{2,t}(n); and (iii) for K_{3,3}-minor-free graphs of order n≥1186, q(G) ≤ q(F_{3,3}(n)) with equality if and only if G=F_{3,3}(n). The proofs combine spectral perturbation arguments with known edge-density and degree-sequence results. The paper also proposes Conjecture 4.4 for general K_{s,t}-minor-free graphs of large order.","tokens_in":12012,"tokens_out":20262,"duration_ms":164230,"significance":"If the missing justifications are supplied, the results constitute a meaningful contribution to spectral extremal graph theory for minor-closed graph classes. Theorems 1.2 and 1.3 are essentially self-contained modulo cited lemmas and give sharp, explicit upper bounds together with unique extremal graphs. The paper makes good use of Perron-Frobenius theory, Rayleigh quotient inequalities, and known edge-density bounds. The main caveat is that Theorem 1.4 depends on four degree-sequence lemmas (Lemmas 2.5–2.8) that are stated in adapted form without proof or a precise reference; if those lemmas are correct, the K_{3,3} result is plausible and well supported otherwise.","major_comments":[{"comment":"The paper states that Lemmas 2.5–2.8 are 'a little different from their original forms [16], but indeed they are correct according to original proofs,' yet it neither proves them nor identifies the exact statements in [16] from which they follow. These lemmas are load-bearing for Theorem 1.4: Lemma 4.2 Case 1 invokes Lemmas 2.5 and 2.6, and Lemma 4.3 Case 1 invokes Lemmas 2.7 and 2.8, to obtain the crucial bound q(G) ≤ n+2. If any of the adapted lemmas fails, the proof cannot force Δ(G)=Δ′(G)=n−1, and the uniqueness of F_{3,3}(n) collapses. The adaptation is nontrivial because [16] concerns planar graphs with e(G) ≤ 3n−6, whereas K_{3,3}-minor-free graphs can have e(G) ≤ 3n−5; the changed constants and ranges are essential to the edge-count inequalities in Lemmas 4.2 and 4.3. The authors must provide full proofs of the adapted lemmas or a detailed derivation from the original results.","section":"Section 2 (Lemmas 2.5–2.8)"},{"comment":"The proofs repeatedly assert 'Clearly G′ is K2,3-minor free' or 'Clearly G′ is K3,3-minor free' after adding edges to path endpoints or performing edge switches (e.g., Theorem 1.2 Cases 1–3, Theorem 1.4 Cases 1–3). These assertions are load-bearing, because the contradiction relies on G′ being minor-free and having a larger signless Laplacian spectral radius. The claims are likely correct, but they require at least a short argument showing that the described operations cannot create the forbidden minor from the path/triangle structure. The authors should supply these justifications rather than leaving them to the reader.","section":"Section 4 (Theorem 1.4 Claim, Cases 1–3; also Theorem 1.2 Claim)"}],"minor_comments":[{"comment":"The text contains a typo: 'Perron–Fronbenius' should be 'Perron–Frobenius'.","section":"Proof of Theorem 1.4"},{"comment":"There are minor language errors: 'Furtherer' in Theorem 1.2 should be 'Further', and 'combing' in Theorem 1.3 should be 'combining'.","section":"Proofs of Theorems 1.2 and 1.3"},{"comment":"In the displayed inequality, the parentheses are unbalanced: the term should read '(e(G) − d(u) − e(N(u)))' with a closing parenthesis before the equal sign.","section":"Lemma 4.1"},{"comment":"The notation 'n/6 + 1 ≤ d_{k+1} ≤ · · · ≤ d_2 ≤ n − 61' is confusing because degree sequences are nonincreasing; rewriting the hypotheses as d_2 ≤ n−61 and d_{k+1} ≥ n/6+1 would improve readability.","section":"Lemmas 2.5 and 2.7"}],"recommendation":"major_revision","confidential_remarks":"The central gap is the unproved adaptation of Lemmas 2.5–2.8 from the planar-graph setting of [16]. Because the constants differ in a way that is essential to the later argument, this is not a routine citation issue and needs to be resolved with full proofs or precise theorem-level references. The 'clearly minor-free' steps are secondary but should be cleaned up. The results are likely correct and of interest to the spectral extremal graph theory community, so revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives the first sharp signless Laplacian spectral radius bounds for K2,3-minor-free and K3,3-minor-free graphs of large order, with the extremal graphs identified. Theorems 1.2 and 1.3 are convincing and largely self-contained. Theorem 1.4 is plausible but propped up by four lemmas imported from a planar graph paper, stated in modified form, with no proof or exact reference.\n\nWhat is genuinely new: the equality characterizations for K2,t-minor-free graphs, including the t=3 case with the explicit polynomial for q(F2,3(n)), and the unique extremal graph among K3,3-minor-free graphs. The proof of Theorem 1.2 is a clean structural argument: Lemma 2.1 forces a universal vertex, then G−u is a union of triangles plus at most one short path. The switching claims are terse but check out. Theorem 1.3 is a neat reduction to the existing K2,t-free result, with a short argument that the (t−1)-regular extremal graph must decompose into K_t's.\n\nThe soft spot is real and load-bearing for Theorem 1.4. Lemmas 2.5–2.8 are introduced as \"a little different from their original forms\" in [16], which is about planar graphs with edge bound 3n−6; here the edge bound is 3n−5 and the constants have been adjusted accordingly. These lemmas are used exactly where the proof must rule out Δ′ ≤ n−2, so without them the second universal vertex is not forced. The authors assert the modifications are correct \"according to original proofs,\" but a referee cannot verify that without a theorem-by-theorem reconciliation. This is a genuine gap, not a minor citation issue.\n\nThere is also a small slip in Lemma 4.2, Subcase 2.2: if v and v2 are both outside N(v1) and adjacent, they get a third common neighbour, so the bound should be n+3 rather than n+2. It does not change the conclusion—the cap of 65 still holds—but it should be corrected.\n\nBottom line: the K2,3 and general K2,t results are solid and worth having; the K3,3 result is likely true but not fully verified as written. This deserves a serious referee: send it out with a request to prove or precisely cite the modified lemmas.","headline":"Sharp signless Laplacian bounds for K2,3- and K3,3-minor-free graphs, with the K3,3 case resting on four unproved adapted lemmas that need proof or exact reference.","tokens_in":12524,"tokens_out":6784,"would_cite":true,"duration_ms":63139,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C35","05C83"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $K_{2,t}$-minor-free graphs of large order, the signless Laplacian spectral radius obeys a sharp closed-form upper bound, with equality exactly on the join graph $F_{2,t}(n)$; the $K_{3,3}$-minor-free case is also settled for $n\\ge…","keywords":["signless Laplacian spectral radius","minor-free graphs","extremal graph theory","K2,t-minor free graph","K3,3-minor free graph","Q-index","Perron-Frobenius theorem","graph eigenvalues"],"falsifier":"Enumerate all degree sequences satisfying the hypotheses of Lemmas 2.5–2.8 near the thresholds and compute the signless Laplacian spectral radius; a single sequence with $q(G)>n+2$ would break the key bound that forces the $K_{3,3}$ extremal graph to contain two universal vertices. Alternatively, build an edge-maximal $K_{3,3}$-minor-free graph with two universal vertices and one remaining path of length 3; the proof's switching argument predicts it cannot be extremal, so any such graph with $q$ exceeding $q(F_{3,3}(n))$ would refute Theorem 1.4.","tokens_in":11465,"feed_emoji":"📈","tokens_out":10882,"duration_ms":97017,"temperature":0.7,"pith_summary":"The paper identifies the exact graphs that maximize the signless Laplacian spectral radius, the largest eigenvalue of $D(G)+A(G)$, among graphs with no $K_{2,t}$ minor and, for large order, no $K_{3,3}$ minor. It proves a closed-form upper bound and shows that equality holds exactly on the join family $F_{s,t}(n)=K_{s-1}\\vee(p\\cdot K_t\\cup K_r)$ under a residue condition; in the $K_{2,3}$ case the maximizer is unique for $n\\ge22$, and in the $K_{3,3}$ case for $n\\ge1186$. This matters because it turns a forbidden-minor structural condition into a sharp spectral inequality, giving the first signless Laplacian extremal results for these minor classes beyond the previously known $K_{2,2}$ case.","feed_headline":"A sharp signless-Laplacian bound for K2,t-minor-free graphs","feed_subtitle":"Equality holds only on the join graph F2,t(n), and the K3,3-minor-free case is settled for order at least 1186.","key_machinery":"The load-bearing family is $F_{s,t}(n)=K_{s-1}\\vee(p\\cdot K_t\\cup K_r)$, where $n-s+1=pt+r$ and $0\\le r<t$: a clique of $s-1$ universal vertices joined to disjoint copies of $K_t$ and one $K_r$. The proof first shows that a spectral extremal graph must be edge-maximal, then uses Perron–Frobenius and a degree-weighted inequality to force one universal vertex for $K_{2,t}$ or two universal vertices for $K_{3,3}$. Once those universal vertices exist, the remaining graph must consist of disjoint triangles and at most a path of length 1 or 2, and the paper proves by explicit edge switches that any longer path would raise $q(G)$, contradicting maximality. Four adapted degree-sequence lemmas carry the delicate step of capping $q$ at $n+2$ for graphs with near-universal but not universal vertices.","core_discovery":"On its own terms, the discovery is an extremal classification. If $G$ has no $K_{2,t}$ minor and order $n\\ge t^2+4t+1$ with $t\\ge3$, then $q(G)\\le \\frac{n+2t-2+\\sqrt{(n-2t+2)^2+8t-8}}{2}$, with equality precisely when $n\\equiv1\\pmod t$ and $G=F_{2,t}(n)$. For $t=3$ this sharpens to the statement that for $n\\ge22$, $F_{2,3}(n)$ is the unique maximizer. For $K_{3,3}$-minor-free graphs, the paper shows that for $n\\ge1186$ the unique maximizer is $F_{3,3}(n)$, with $q(F_{3,3}(n))$ the largest root of the displayed cubic. The paper also states what it calls Conjecture 4.4: for $2\\le s\\le t$ and sufficiently large $n$, $F_{s,t}(n)$ should be the unique maximizer among $K_{s,t}$-minor-free graphs.","pith_inferences":["The residue condition $n\\equiv1\\pmod t$ looks like a technical artifact: for orders in other residue classes the stated bound is not attained, and the natural prediction is that $F_{2,t}(n)$ still maximizes $q$, merely with a value strictly below the bound.","The explicit edge-switch arguments suggest a general smoothing principle for the signless Laplacian: in spectral extremal minor-free graphs, long pendant paths are unstable, so extremal graphs should always be highly clustered unions of cliques around a small universal core.","The four adapted degree-sequence lemmas are the fragile hinge of the $K_{3,3}$ proof; checking them computationally on degree sequences near $n=1186$ would be a cheap way to test whether the structural conclusion is sound.","If the same strategy is pushed to general $K_{s,t}$, the thresholds should grow polynomially in $t$ rather than exponentially, and the current bounds $t^2+4t+1$ and $1186$ are the natural starting points."],"forward_implications":["For $K_{2,3}$-minor-free graphs with $n\\ge22$, the maximum signless Laplacian spectral radius is exactly $q(F_{2,3}(n))$, the largest root of the cubic in Lemma 2.3(ii), and $F_{2,3}(n)$ is the unique extremal graph.","For $t\\ge4$, the upper bound is tight exactly when $n\\equiv1\\pmod t$; in other residue classes the inequality is strict, so the theorem leaves the exact maximum open for those orders.","For $K_{3,3}$-minor-free graphs of order $n\\ge1186$, $F_{3,3}(n)$ is the unique extremal graph, completely settling the $Q$-spectral extremal problem for this minor class at large orders.","The extremal graphs in all settled cases are edge-maximal and contain one or two universal vertices, so any future counterexample would have to avoid that structure.","The paper's Conjecture 4.4 says the same join family $F_{s,t}(n)$ should maximize $q$ among all $K_{s,t}$-minor-free graphs for $2\\le s\\le t$ and sufficiently large $n$."],"supporting_citations":[{"why":"Supplies Lemmas 2.5–2.8, the adapted degree-sequence bounds $q(G)\\le n+2$ that force an extremal $K_{3,3}$-minor-free graph to have two universal vertices.","marker":"[16]"},{"why":"Supplies Lemmas 2.1 and 2.2, the $K_{2,t}$-free signless Laplacian bounds used to force one universal vertex and to state the sharp $K_{2,t}$ inequality.","marker":"[8]"},{"why":"Supplies the edge bound $e(G)\\le3n-5$ and edge-maximal degree facts used in Lemmas 4.2 and 4.3 for $K_{3,3}$-minor-free graphs.","marker":"[6]"},{"why":"Supplies the edge bound for $K_{1,t}$-minor-free graphs used in Theorem 1.3 to rule out non-clique regular components.","marker":"[5]"},{"why":"Supplies the degree-weighted neighborhood inequality $q(G)\\le\\max_v(d(v)+\\frac1{d(v)}\\sum_{w\\in N(v)}d(w))$ used in Lemma 4.1.","marker":"[10]"},{"why":"Supplies the nonnegative-matrix comparison criterion used in Lemmas 4.2 and 4.3 to cap $q(G)$ at $n+2$.","marker":"[2]"}],"fun_headline_variants":["Sharp q(G) bound for K2,t-minor-free: equality on F2,t(n)","Unique K3,3-minor-free extremal for n≥1186","Tight signless Laplacian: K2,t-minor-free, iff F2,t(n)","Conjecture: F_{s,t}(n) maximizes q among K_{s,t}-minor-free","Uniqueness of extremal K2,3-minor-free graph for n≥22"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $K_{3,3}$ proof depends on four degree-sequence lemmas taken from earlier work in slightly changed forms; the paper says these altered forms are correct but does not prove them, and if any one fails, the conclusion that the extremal graph must have two universal vertices collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sharp q(G) bound for K2,t-minor-free: equality on F2,t(n)","Unique K3,3-minor-free extremal for n≥1186","Tight signless Laplacian: K2,t-minor-free, iff F2,t(n)","Conjecture: F_{s,t}(n) maximizes q among K_{s,t}-minor-free","Uniqueness of extremal K2,3-minor-free graph for n≥22"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000313,"raw_usage":{"total_tokens":1836,"prompt_tokens":1060,"completion_tokens":776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":657}},"tokens_in":676,"tokens_out":776,"duration_ms":6832,"temperature":1.0,"reasoning_tokens":657,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:50:15.558826+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all degree sequences satisfying the hypotheses of Lemmas 2.5–2.8 near the thresholds and compute the signless Laplacian spectral radius; a single sequence with $q(G)>n+2$ would break the key bound that forces the $K_{3,3}$ extremal graph to contain two universal vertices. Alternatively, build an edge-maximal $K_{3,3}$-minor-free graph with two universal vertices and one remaining path of length 3; the proof's switching argument predicts it cannot be extremal, so any such graph with $q$ exceeding $q(F_{3,3}(n))$ would refute Theorem 1.4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemmas 2.5–2.8, the adapted degree-sequence bounds $q(G)\\le n+2$ that force an extremal $K_{3,3}$-minor-free graph to have two universal vertices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemmas 2.1 and 2.2, the $K_{2,t}$-free signless Laplacian bounds used to force one universal vertex and to state the sharp $K_{2,t}$ inequality."},{"cited_title":"Fang, Bounds of eigenvalues of K3,3-minor free graphs, J","cited_arxiv_id":null,"evidence_quote":"Supplies the edge bound $e(G)\\le3n-5$ and edge-maximal degree facts used in Lemmas 4.2 and 4.3 for $K_{3,3}$-minor-free graphs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the edge bound for $K_{1,t}$-minor-free graphs used in Theorem 1.3 to rule out non-clique regular components."},{"cited_title":"Merris, A note on Laplacian graph eigenvalues, Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Supplies the degree-weighted neighborhood inequality $q(G)\\le\\max_v(d(v)+\\frac1{d(v)}\\sum_{w\\in N(v)}d(w))$ used in Lemma 4.1."},{"cited_title":"Berman, R","cited_arxiv_id":null,"evidence_quote":"Supplies the nonnegative-matrix comparison criterion used in Lemmas 4.2 and 4.3 to cap $q(G)$ at $n+2$."}],"review_version":1}