{"id":"4a7fb373-322b-4b1f-b398-917844272104","arxiv_id":"2607.15071","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any non-complete-bipartite graph with m edges and spectral radius at least √m has a book of size at least ρ(G)/3 and at least ρ(G) triangular edges.","lead":"This paper proves two sharp bounds for graphs whose spectral radius is at least the square root of their edge count: the largest book has size at least one third of the spectral radius, and the graph contains at least that many triangular edges. It resolves an open problem of Li, Liu and Zhang and a conjecture of Li, Feng and Peng about Nosal graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the delicate Claim 1, though terse, is valid after filling in the classification of V*","rationale":"The reader identified Claim 1 as the most delicate step, which I agree is the right place to focus. However, after a detailed reconstruction, the claim is correct and the subsequent vector comparison is sound. The only issue is a notational typo in the definition of p_z, which is easily inferred from context and does not affect the mathematics. Therefore, the manuscript's central results—Theorems 1.10 and 1.11—are supported, and the original ACCEPT verdict should stand.","tokens_in":13539,"tokens_out":20468,"duration_ms":180085,"concrete_test":"Formally verify the classification V* = {u*} ∪ {y'} ∪ N_U(y') ∪ (N_W(y) ∩ N_W(y')) in the equality case of Lemma 4.3 and check that every possible edge among these vertices is triangular (using u*, y, or y' as a common neighbor). This fills the terse step in Claim 1 and would expose any overlooked edge class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the main theorems and the reader's flagged weakest step, Claim 1 in Subcase 2.2 of Theorem 1.11. The manuscript's proof of Claim 1 is compressed, but the step is recoverable: in the equality case of Lemma 4.3, the set of triangular edges is exactly S(y,y'), so V* = {u*} ∪ {y'} ∪ N_U(y') ∪ (N_W(y) ∩ N_W(y')). Every edge in G[U] is triangular via u*, every edge from u* to U is triangular via any common U-neighbor, and every edge from a W-vertex in V* to any U-vertex or other W-vertex is triangular via y or y'. Thus G[V*] = H and N_G(v) ∩ V* is independent for v outside V*, as claimed. I also checked the vector-comparison argument: the apparent typo in the definition of p_z (both subscripts showing 'Vz' rather than one overline) is resolved by context—the intended definition is d_{V_z}(v) + 1/2 d_{\\overline{V_z}}(v) for v ∈ \\overline{V_z}, which makes j^T p_z = ρ|V_z| + e(H) hold. No fatal gaps, fitted parameters, or circular reasoning found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies m-edge graphs G with no isolated vertices, spectral radius ρ(G) ≥ √m, and G not isomorphic to a complete bipartite graph. Theorem 1.10 proves bk(G) ≥ ρ(G)/3, and Theorem 1.11 proves τ(G) ≥ ρ(G). These imply, for Nosal graphs, bk(G) > √m/3 and τ(G) > √m, resolving Problem 1.6 of Li–Liu–Zhang and Conjecture 1.3 of Li–Feng–Peng. The proofs use Perron-vector weighting, double counting over triangles (Section 3), and a localization around a maximum eigenvector entry with a tight-case structural analysis and a vector comparison argument (Section 4).","tokens_in":13887,"tokens_out":34585,"duration_ms":325796,"significance":"The paper settles two open problems in spectral extremal graph theory and gives essentially optimal constants up to o(√m) slack. The bounds are stated in the stronger form involving ρ(G) rather than only √m, and this stronger form is essential to the proofs. The derivations are self-contained and direct: they use only Perron–Frobenius, standard eigenvalue identities, and careful counting, with no fitted parameters and no circularity. The tight-case analysis in Theorem 1.11, especially Claim 1, is delicate but valid after filling in the compressed details. The optimality examples in Section 5 are also useful, modulo a minor numerical typo noted below.","major_comments":[],"minor_comments":[{"comment":"The displayed equality τ(H) = 2t+1 = √e(H) + t is false for all t ≥ 1. Indeed e(H)=4t^2+3t+1 gives √e(H) = 2t + 3/4 + O(1/t), so τ(H) = √e(H) + 1/4 + o(1). The asymptotic optimality of the constant 1 still follows from τ(H)/√m → 1, but the stated equality should be corrected.","section":"§5, Further work"},{"comment":"There are missing overlines in the definition of p_z and in several equations of Claim 2. The intended definition is p_{v,z} = d_{V_z}(v) + 1/2 d_{\\overline{V_z}}(v) for v ∈ \\overline{V_z}; with the printed notation the identity j^T p_z = ρ|V_z| + e(H) does not hold. Please correct the notation throughout Claim 2.","section":"§4, Subcase 2.2, definition of p_z"},{"comment":"The proof of Claim 1 is very compressed. In particular, the identification of V* and the explicit description of E(H) are asserted with \"Clearly\"; the equality analysis in (8) that rules out additional triangular edges (e.g., edges within N_U(y') or within N_W(y)∩N_W(y')) should be spelled out. The claim is valid, but this is a delicate structural step that needs a fuller justification for the reader.","section":"§4, Claim 1"},{"comment":"The step \"Thus x_v = 1/2 for any v ∈ U_1\" is terse. It should be explained: if some x_v > 1/2, then a B-neighbor u of v would have β_u ≥ x_v > 1/2, forcing u ∈ U_1 \\ U_2 and contradicting U_2 = U_1. The argument is sound but not immediate.","section":"§4, Case 1"},{"comment":"The renormalization of the Perron vector from max=1 to sum=1 should be flagged explicitly. Earlier sets B, U_1, U_2 were defined using the max=1 normalization; the final argument uses a different scaling. Since the graph-theoretic objects V*, V_z, p_z do not depend on this scaling, the proof is valid, but the transition should be clarified to avoid confusion.","section":"§4, after Claim 2"},{"comment":"The sentence \"Then G is a complete bipartite graph by the definition of z_e\" is terse. A brief justification (fix an edge uv, partition V into N(u) and N(v), and use z_e=0 to force all cross edges) would improve readability.","section":"§3, Lemma 3.2"}],"recommendation":"minor_revision","confidential_remarks":"The main theorems are sound, significant, and directly resolve the stated open problems. I have no reservations about the central claims. The manuscript needs only local corrections: fix the false equality in Section 5, repair the missing overlines in Claim 2, and expand the proof of Claim 1. These do not affect the validity of the main results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a clean, genuinely new result. It answers Problem 1.6 of Li–Liu–Zhang and confirms Conjecture 1.3 of Li–Feng–Peng, improving the book-size constant from 1/4 to the optimal 1/3 and proving τ(G) ≥ ρ(G). The proof strategy extends the Perron-vector weighting of Zhai, Li and Lou—not revolutionary, but the execution is solid. Theorem 1.10's double-counting argument is short and correct: Lemma 3.1's identity checks, Lemma 3.3's lower bound is fine, and the final inequality gives bk ≥ ρ/3. Theorem 1.11 is more intricate. The localization identity (5) is exactly right, and Lemma 4.3's edge counting gives the right R. The proof branches into two cases. Case 1 is a straightforward regularity argument. Case 2 is the heart: under τ(G) < ρ, derive a contradiction. The vector comparison in Claim 2 is sound, and the Neumann series step is legitimate because ρ(H) < ρ. I checked the equality case in Subcase 2.2: the proof of Claim 1 is terse, but the stress-test's reconstruction works—once equality holds in Lemma 4.3, the classification of V* follows, and the neighborhood independence is direct. The only real blemish is notation: the definition of p_z has a typo (missing overline) and some subscripts render ambiguously. That's a referee fix, not a substantive flaw.\n\nOne minor thing: Lemma 3.2's proof that K_3(G) is nonempty is a bit quick; it hinges on z_e = 0 for every edge implying complete bipartite. It's correct, but a reader might want one more sentence. Also, the abstract's promise of 'no isolated vertices' is important—the complete bipartite exception needs it—and the authors handle the disconnected case properly.\n\nBottom line: the results are real, the constants are optimal (the examples in Section 5 confirm this), and the proofs are internally consistent. This deserves a serious referee. I'd accept it after a round of polishing.","headline":"Settles two open problems on Nosal graphs with optimal constants; proofs check out, with a few terse spots that need polishing.","tokens_in":14370,"tokens_out":2194,"would_cite":true,"duration_ms":23181,"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-edge graphs with no isolated vertices, spectral radius ≥ √m forces a book of size ρ/3 and ρ triangular edges.","keywords":["booksize","triangular edges","spectral radius","Nosal graph","spectral extremal graph theory","supersaturation","Perron vector","books"],"falsifier":"Construct an m-edge graph with no isolated vertices, ρ ≥ √m, and not complete bipartite, with bk(G) < ρ/3 or τ(G) < ρ. More specifically, for Theorem 1.11, search for a graph where equality holds in the local count of Lemma 4.3 but some outside vertex's neighborhood in V* contains a triangular edge of H; such a graph would refute Claim 1 and invalidate the vector comparison.","tokens_in":13469,"feed_emoji":"📚","tokens_out":6800,"duration_ms":62443,"temperature":0.7,"pith_summary":"This paper proves that any m-edge graph with no isolated vertices whose spectral radius ρ is at least √m, and which is not complete bipartite, must contain a book of size at least ρ/3 and at least ρ edges that lie in triangles. The result answers a question about the optimal constant for book size in Nosal graphs (graphs with ρ(G) > √m) and confirms a conjecture about triangular edges. It implies that every Nosal graph has a book of size greater than √m/3 and more than √m triangular edges. The proof is a spectral double-counting argument that uses the Perron vector to turn the global condition ρ ≥ √m into local triangle counts.","feed_headline":"Spectral radius forces books of size ρ/3","feed_subtitle":"Graphs with ρ ≥ √m also have ρ triangular edges, unless they are complete bipartite.","key_machinery":"The proof centers on the Perron vector x of G: after normalizing its largest entry to 1 and picking a vertex u* where it is attained, it expresses ρ² - m as a sum of excess contributions from edges inside N(u*) minus deficit terms f(w) for outside vertices. A local count shows that any 'good' edge uv forces at least R(d_U(u), d_U(v), c_uv) triangular edges, and the tight case is analyzed through the subgraph H of triangular edges and the matrix M = ρI - A(H) with a nonnegative inverse obtained by the Neumann series. The final step is a vector comparison showing that if τ(G) < ρ, the deficit strictly exceeds the excess, contradicting ρ ≥ √m.","core_discovery":"The central discovery is a pair of sharp bounds: for every m-edge graph G with no isolated vertices, ρ(G) ≥ √m, and G not complete bipartite, the maximum book size (the largest number of triangles sharing a common edge) satisfies bk(G) ≥ ρ(G)/3, and the number of triangular edges satisfies τ(G) ≥ ρ(G). Because ρ(G) ≥ √m, these are exactly the optimal constants 1/3 and 1 in the conjectured lower bounds for Nosal graphs (ρ(G) > √m). The paper also constructs examples showing that neither constant can be improved.","pith_inferences":["The localization method around a Perron-maximal vertex could be adapted to find optimal constants for generalized books B_{s,k}, since the same excess-vs-deficit structure may appear there.","The structurally rigid tight case (Claim 1) suggests a general principle: in spectral supersaturation, equality in local counts forces strong induced-structure constraints; probing this rigidity may unify other tight cases.","A direct test would be to check whether the assumption 'no isolated vertices' can be relaxed: the theorems likely hold for the non-isolated core of a graph, with isolated vertices appended without effect."],"forward_implications":["Every Nosal graph has a book of size greater than √m/3 and more than √m triangular edges.","The constants 1/3 and 1 are best possible: the paper's examples show no larger factor can hold uniformly.","The stronger bounds are stated in terms of ρ(G) rather than √m, so the result is a genuine spectral supersaturation theorem, not just a corollary of the √m threshold.","For t ≥ 4, the same supersaturation phenomenon extends to edges contained in K_t, with a linear lower bound (Theorem 5.4)."],"fun_headline_variants":["ρ ≥ √m forces book size ≥ ρ/3","Sharp bounds: bk ≥ ρ/3, τ ≥ ρ, unless complete bipartite","Spectral radius dictates book size and triangular edges","Optimal constants: bk ≥ ρ/3 and τ ≥ ρ from ρ ≥ √m"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 1.11 rests on Claim 1, which asserts that when equality holds in the local triangular-edge count, the triangular-edge subgraph is induced and every outside vertex sees an independent set inside it; if that structural rigidity misses an equality case, the vector comparison that produces the contradiction loses its foundation.","fun_headline_variants_meta":{"raw":{"variants":["ρ ≥ √m forces book size ≥ ρ/3","Sharp bounds: bk ≥ ρ/3, τ ≥ ρ, unless complete bipartite","Spectral radius dictates book size and triangular edges","Optimal constants: bk ≥ ρ/3 and τ ≥ ρ from ρ ≥ √m"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000421,"raw_usage":{"total_tokens":2027,"prompt_tokens":794,"completion_tokens":1233,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1155}},"tokens_in":538,"tokens_out":1233,"duration_ms":11853,"temperature":1.0,"reasoning_tokens":1155,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T00:15:45.645794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an m-edge graph with no isolated vertices, ρ ≥ √m, and not complete bipartite, with bk(G) < ρ/3 or τ(G) < ρ. More specifically, for Theorem 1.11, search for a graph where equality holds in the local count of Lemma 4.3 but some outside vertex's neighborhood in V* contains a triangular edge of H; such a graph would refute Claim 1 and invalidate the vector comparison.","supporting_citations":[],"review_version":1}