{"id":"d3612619-f9ac-42ce-b5cc-6a2941f59e41","arxiv_id":"2411.19110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Among gem-free graphs with m edges, m odd and m at least 23, excluding the standard extremal graph, the spectral radius is maximized only by the graph S^2_{(m+5)/2,2}.","lead":"This paper proves a sharp upper bound on the largest eigenvalue of gem-free graphs with an odd number of edges, after the usual best graph is taken out of consideration. The result identifies the exact runner-up graph, completing the second-extremal case for this forbidden-pattern problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Construction 2's claimed contradiction for odd-degree W vertices rests on an unproved sign assertion: the term x_û z_{k0} - x_w z_{v_d} is not implied by x_w < x_v ≤ x_û and may be negative under the stated inequalities.","rationale":"The reader identified the unexpanded 'H5-free after edge move' assertions as the weakest point. My reading agrees the proof is conditional, but the single most load-bearing gap I see is different: the algebraic positivity claim at the end of Construction 2. The paper says 'Combining the fact x_{w_i} < x_v ≤ x_û' the displayed odd-case expression is positive, yet the third term compares x_û against x_w times z_{v_d}/z_{k0}, and z_{v_d}/z_{k0} is strictly larger than 1 because v_d is adjacent to both hubs while k0 is only adjacent to û. The displayed inequalities alone do not imply the needed comparison. If the odd case cannot be repaired, the proof of W = ∅ fails and the theorem is unproved. If it can be repaired by a short eigenvector estimate, the theorem may well be correct; this is exactly a conditional, not a rejection. I therefore keep the reader's CONDITIONAL verdict, but I do not regard the H5-free preservation step as the principal vulnerability.","tokens_in":9827,"tokens_out":39469,"duration_ms":363236,"concrete_test":"Set up the exact Perron equations for G and G'_c in the odd case d_i = 2a+1, and check the sign of f_{w_i} over all parameter choices satisfying the inequalities used in Section 3, especially x_w < x_v - x_û/ρ, x_v ≤ x_û, e(W)=0, and ρ > 5.1. Run a numerical/computer-algebra scan for a = 1,2 and m = 23..200; if any valid configuration gives f ≤ 0, the contradiction in Claim 3.7 collapses, while if all give f > 0, extract the missing eigenvector estimate and verify it formally.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of W = ∅ ends by constructing G'_c from G_c and showing ρ(G'_c) > ρ(G_c), reduced to f_{w_i} > 0 for each replaced vertex w_i. For odd d_i, f_{w_i} is written as three groups and asserted to be positive because x_{w_i} < x_v ≤ x_û. This implication is not valid: z_{k_i^0} is the Perron coordinate of a pendant vertex adjacent only to û, while z_{v_{d_i}} is the coordinate of a leaf adjacent to both û and v, so z_{v_{d_i}} > z_{k_i^0}. The cited bounds give x_û - x_{w_i} > x_û/ρ, but they do not dominate x_{w_i}(z_{v_{d_i}}/z_{k_i^0} - 1). For example, with ρ ≈ 5.2, x_v ≈ x_û, x_{w_i}/x_û ≈ 0.9, and z_v/z_û ≈ 1, the third term is negative, and the proof supplies no additional eigenvector estimate excluding such parameter regimes. Since this positivity is exactly what produces the contradiction W = ∅ and hence the extremal characterization, the central inequality is not established.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Brualdi-Hoffman spectral Turan problem for gem-free (H5-free) graphs with a fixed number m of edges and no isolated vertices. Building on the known theorem that S_{(m+3)/2,2} is the unique spectral extremal graph for all H5-free graphs of odd size, the authors determine the runner-up for odd m at least 23: they claim that among H5-free graphs not isomorphic to S_{(m+3)/2,2}, the maximum spectral radius is attained uniquely by S^2_{(m+5)/2,2}. The proof fixes an extremal graph G_hat and a Perron vertex u_hat, classifies the components of G_hat[N(u_hat)] as triangles or stars, derives inequalities for e(W), and then uses a sequence of local edge moves and two graph constructions to force W = V(G_hat) \\ N[u_hat] to be empty. Once W is empty, the structure of the graph is identified with S^t_{(m+t+3)/2,2}, and an eigenvalue comparison from Lemma 2.2 forces t=2.","tokens_in":10120,"tokens_out":28161,"duration_ms":233418,"significance":"If correct, Theorem 1.2 is a natural and sharp stability-type refinement of the gem-free spectral theorem of Zhang-Wang and Yu-Li-Peng, parallel to the C5/C6 result of Sun-Li-Wei. The paper contains no fitted parameters, the candidate extremal graphs are explicit, and the broad strategy is recognizable and plausible. However, several load-bearing steps are currently asserted rather than proved, and one algebraic positivity claim in the construction phase is not justified. These gaps concern the main contradiction that excludes W nonempty, so they are central rather than cosmetic.","major_comments":[{"comment":"For odd d_i, the quantity f_{w_i} is expanded as the sum of two nonnegative-looking groups plus the term (x_hat_u z_{k_i^0} - x_{w_i} z_{v_{d_i}}), and this last term is asserted to be positive from x_{w_i}<x_v<=x_hat_u. This implication is invalid. In G'_c, k_i^0 is a pendant vertex adjacent only to u_hat, while v_{d_i} is a common leaf adjacent to both u_hat and v, so z_{v_{d_i}}=(z_u_hat+z_v)/rho > z_u_hat/rho = z_{k_i^0}. The stated inequalities allow parameter regimes, e.g. x_v close to x_hat_u and x_{w_i} close to x_hat_u with z_v close to z_u_hat, in which the last term is negative and can dominate the first two groups. Since the proof of rho(G'_c)>rho(G_c) depends on f_{w_i}>0 for every i, and this inequality is exactly what rules out W nonempty, the contradiction in Lemma 3.7 is not established.","section":"Section 3, Lemma 3.7, Construction 2, odd d_i case"},{"comment":"The assertion that G*_c is isomorphic to S^t_{(m+t+3)/2,2} with t>=1 is not justified. If N0(u_hat)=empty and every d_i is even, the construction deletes all vertices of W and adds no pendant vertices, so G*_c is isomorphic to S_{(m+3)/2,2}, not to a graph with t>=1. Such configurations are H5-free and satisfy the standing hypotheses; for example, take u_hat and v joined to r leaves and one vertex w joined to four of those leaves, with m=2r+5 and r>=9. In this case Lemma 2.2(ii), which requires even t>=4, cannot be applied, and the final contradiction does not cover the case.","section":"Section 3, Lemma 3.7, after Construction 2"},{"comment":"Several edge moves are asserted to preserve H5-freeness without proof: Lemma 3.4(ii) says 'Then G' is still H5-free', Claim 3.1 says 'Clearly, G* is H5-free', and Claim 3.2 says 'It is checked that G5 is still an H5-free graph'. These assertions are load-bearing because Lemma 2.1 can only be invoked when the moved graph remains in the admissible family G(m,H5). In particular, at the moment of Claim 3.1, e(W) is only known to be at most 1 from equation (4), and a vertex w may still have neighbors outside the triangle {z1,z2,z3}; the 'clearly' assertion needs a detailed case analysis. Please supply complete proofs of H5-freeness for these moves or replace them by operations whose H5-freeness is directly verified.","section":"Section 3, Lemma 3.4(ii), Claim 3.1, Claim 3.2"}],"minor_comments":[{"comment":"In the displayed formula for f_{w_i} when d_i is odd, the index v_{j+d_i/2} is not an integer; the intended index is likely v_{j+(d_i+1)/2} or an equivalent relabeling.","section":"Section 3, Construction 2, odd d_i formula"},{"comment":"The inequality e(W) < e(N+(u_hat)) - |N+(u_hat)| + 2 - sum_{u in N0(u_hat)} x_u/x_u_hat is equation (4), not equation (3); the citation should be corrected.","section":"Lemma 3.6(ii)"},{"comment":"The text says 'Li, Zhou and Zou completely solved Conjecture 1.1', but reference [6] is attributed to 'S.C. Li, S.S. Zhao, L.T. Zou'; the author names should be harmonized.","section":"Introduction, reference to [6]"},{"comment":"The abstract states that every gem-free graph G with m edges satisfies the bound, while the formal statement of Theorem 1.1 in the introduction requires m>=11; the abstract should include this condition.","section":"Abstract and Theorem 1.1"},{"comment":"The phrase 'if G in G(m,H5) \\setminus S_{(m+3)/2,2} be a graph of odd size m>=23' should read 'is a graph'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the problem is sensible and the claimed result is likely true, but the written proof of Lemma 3.7 has substantive gaps. The positivity of f_{w_i} for odd d_i requires an additional eigenvector estimate, the t=0 case of the constructed graph is missing, and the H5-freeness of several local moves is asserted without proof. These are not merely presentational issues; they require new arguments. I recommend major revision rather than rejection because the overall strategy is credible and the gaps may be repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The theorem is new and the paper is a legitimate extension of the Sun-Li-Wei C5/C6 approach. Zhang-Wang and Yu-Li-Peng only had the first extremal graph; this paper identifies the runner-up for odd m. The structural setup — components of N(u) are triangles or stars, e(W) is small, at most one star component — is clean and mostly checkable. The Perron-vector inequalities in Section 3 are standard.\n\nThe problem is in Lemma 3.7, Construction 2, odd-degree case. The proof asserts each term in f_{w_i} is positive because x_w < x_v ≤ x_û. For the last term, x_û z_{k_i^0} − x_{w_i} z_{v_{d_i}}, that implication does not follow. z_{k_i^0} is the coordinate of a pendant vertex adjacent only to û, while z_{v_{d_i}} is a leaf adjacent to both û and v, so z_{v_{d_i}} > z_{k_i^0}. The ratio z_{v_d}/z_{k^0} can exceed x_û/x_w; the stated inequalities do not dominate it. The paper gives no additional estimate for z_v/z_û, so the claimed positivity is not established. Since this positivity drives the contradiction W = ∅, the extremal characterization is not proven as written. It is a gap, not necessarily a false theorem — the combined sum may still be positive — but the proof as written does not show it.\n\nSecondary issue: several H5-freeness assertions after edge moves are declared rather than shown ('Clearly', 'It is checked'). Some are routine, but because a single overlooked gem breaks the extremal contradiction, a referee would need those cases expanded.\n\nFor whom: specialists in spectral extremal graph theory. The result is plausible and fits the pattern of the C5/C6 papers; if the gap is repaired, it is a solid contribution. I would send it to review because the theorem is meaningful and the framework is sound, but the revision needs to fill the odd-degree estimate and spell out the free-ness checks. My own verdict is skeptical pending that fix.","headline":"New second-extremal result for gem-free graphs, but the proof of the odd-degree case in Construction 2 rests on an unproved eigenvector inequality.","tokens_in":10647,"tokens_out":5076,"would_cite":false,"duration_ms":47940,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for odd $m\\ge23$, the spectral radius of every gem-free graph with $m$ edges that is not the extremal graph $S_{\\frac{m+3}{2},2}$ is at most $\\rho(S^2_{\\frac{m+5}{2},2})$, with equality only for…","keywords":["spectral radius","gem-free graph","fan graph H5","Brualdi-Hoffman-Turan type problem","size of graph","extremal graph","stability","Perron vector"],"falsifier":"Enumerate all gem-free graphs on 23 edges by a backtracking generator that rejects any graph containing $H_5$, and compute their spectral radii. The theorem is false if any graph other than $S_{13,2}$ and $S^2_{14,2}$ has spectral radius greater than $\\rho(S^2_{14,2})$. Equivalently, inspect the intermediate graphs $G'$, $G^{\\star}$, and $G_5$ constructed in Lemma 3.4 and Claims 3.1–3.2: if any of them contains $H_5$ on five vertices, the contradiction argument identifying $S^2_{14,2}$ collapses.","tokens_in":9611,"feed_emoji":"💎","tokens_out":15738,"duration_ms":117313,"temperature":0.7,"pith_summary":"Among graphs with a fixed number $m$ of edges and no copy of the gem $H_5$ — the fan on a 4-vertex path plus a vertex adjacent to every path vertex — the largest spectral radius is known to belong to $S_{\\frac{m+3}{2},2}$, a copy of $K_2$ joined to independent vertices. This paper determines what happens right below that maximum for odd $m\\ge23$: once $S_{\\frac{m+3}{2},2}$ is excluded, every remaining gem-free graph has spectral radius at most $\\rho(S^2_{\\frac{m+5}{2},2})$, and the bound is attained only by that graph. The runner-up $S^2_{\\frac{m+5}{2},2}$ is obtained from $S_{\\frac{m+1}{2},2}$ by attaching two pendant vertices to one of the two hub vertices, so the near-extremal structure is a small modification of the extremal one. The result upgrades the single extremal result to a stability statement: the only way to come close to the spectral record is to move two edges into leaves at one hub.","feed_headline":"Second-best gem-free graph is a K2 with two pendant leaves","feed_subtitle":"For odd m≥23, the runner-up is a K2 hub with two pendant leaves.","key_machinery":"The argument is carried by a Perron-vector edge-shift lemma (Lemma 2.1): if the Perron coordinate of $u$ is at least that of $v$, moving an edge incident to $v$ so that it becomes incident to $u$ strictly increases the spectral radius while preserving the number of edges. The proof repeatedly applies this to a hypothetical extremal graph $\\hat{G}$: each time $\\hat{G}$ deviates from $S^2_{\\frac{m+5}{2},2}$, the lemma produces a gem-free graph with the same number of edges and larger spectral radius, contradicting maximality. To make the contradictions work, two further ingredients are needed: Lemma 3.1 classifies the neighborhood of the extremal vertex (each component is a triangle or a star), and Lemma 2.2 pins down the spectral-radius comparisons, in particular $\\rho(S^2_{\\frac{m+5}{2},2})>\\frac{1+\\sqrt{4m-7}}{2}$, which forces any counterexample to be dense enough that its neighborhood structure falls into the classified cases.","core_discovery":"The central claim is Theorem 1.2. Let $\\mathcal{G}(m,H_5)$ be the family of gem-free graphs with $m$ edges and no isolated vertices. For odd $m\\ge23$, if $G\\in\\mathcal{G}(m,H_5)$ and $G\\not\\cong S_{\\frac{m+3}{2},2}$, then $\\rho(G)\\le\\rho(S^2_{\\frac{m+5}{2},2})$, and equality holds exactly when $G\\cong S^2_{\\frac{m+5}{2},2}$. The runner-up $S^2_{\\frac{m+5}{2},2}$ is the graph formed from $S_{\\frac{m+1}{2},2}$ (a $K_2$ hub joined to $\\frac{m-3}{2}$ isolated vertices) by attaching two pendant vertices to one of the two hub vertices. In other words, once the unique maximum is excluded, the spectral radius over all gem-free graphs of odd size at least 23 is maximized by this explicit split graph with two extra leaves.","pith_inferences":["The Perron-vector edge-shift scheme may transfer to the other fan graphs $H_{2k+1}$ and $H_{2k+2}$, for which the extremal graph is known: the natural guess is that the runner-up is again the extremal split graph with two edges converted into pendant leaves on a dominating vertex, and the inequalities of Lemma 2.2 provide the comparison template.","The threshold $m\\ge23$ comes from inequalities such as $\\rho>5.1$; a computer search over gem-free graphs with $m=11,13,\\ldots,21$ could show whether the runner-up theorem actually holds for smaller odd sizes, and would locate the true threshold.","The proof's repeated 'clearly H5-free' and 'it is checked' steps are the natural targets for machine verification: checking each intermediate edge-switch graph for a copy of $H_5$ would either certify the stability statement or expose a hidden gem that would force a different runner-up."],"forward_implications":["For odd $m\\ge 23$, the spectral extremal problem for gem-free graphs is now resolved up to the runner-up: the maximum is $S_{\\frac{m+3}{2},2}$ and the unique second graph is $S^2_{\\frac{m+5}{2},2}$.","The theorem is a stability statement: any gem-free graph of odd size at least $23$ that is not the known extremal graph has spectral radius strictly below $\\rho(S^2_{\\frac{m+5}{2},2})$ unless it is exactly that graph.","Together with the even-size counterpart mentioned in the introduction, the Brualdi–Hoffman–Turán problem for the gem is settled for all sufficiently large edge numbers, with explicit first and second extremal graphs.","A direct corollary of the proof is that any counterexample would have to contain at least four edges inside the neighborhood of the extremal vertex, so near-extremal graphs are heavily structured around the vertex where the Perron vector attains its maximum."],"supporting_citations":[{"why":"Supplies Lemma 2.1, the Perron-vector edge-shift inequality: moving an edge from a vertex to one with a larger Perron coordinate strictly raises the spectral radius. Every structural contradiction in the proof is built on this move.","marker":"[17]"},{"why":"Supplies Lemma 2.2: the lower bound $\\rho(S^2_{\\frac{m+5}{2},2})>\\frac{1+\\sqrt{4m-7}}{2}$ and the comparison with other $S^t$ graphs decide why the claimed runner-up survives and its rivals do not.","marker":"[13]"},{"why":"Proves the base extremal result (Theorem 1.1) that $\\rho(G)\\le\\rho(S_{\\frac{m+3}{2},2})$ for gem-free graphs of size $m\\ge11$ and formulates the fan-free conjecture; the new theorem is the second-place extension of this result.","marker":"[14]"},{"why":"Independently proves Theorem 1.1 for the gem, establishing the extremal graph $S_{\\frac{m+3}{2},2}$ that the present runner-up result excludes.","marker":"[18]"}],"fun_headline_variants":["Runner-up gem-free graph: K2 hub with two pendant leaves","Odd m≥23: second extremal gem-free graph identified","Spectral max for gem-free graphs: beyond the top graph","For odd m≥23, gem-free runner-up is a K2 plus two leaves","Second-best spectral radius: explicit gem-free graph"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that several local edge switches — reattaching a pendant vertex to another hub, or moving a whole vertex to the universal vertex — never create a copy of the gem $H_5$, and it asserts these checks without expanding them; if any one of these moves secretly produces a gem, the contradiction that forces the runner-up graph fails.","fun_headline_variants_meta":{"raw":{"variants":["Runner-up gem-free graph: K2 hub with two pendant leaves","Odd m≥23: second extremal gem-free graph identified","Spectral max for gem-free graphs: beyond the top graph","For odd m≥23, gem-free runner-up is a K2 plus two leaves","Second-best spectral radius: explicit gem-free graph"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000408,"raw_usage":{"total_tokens":2178,"prompt_tokens":1063,"completion_tokens":1115,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":1026}},"tokens_in":679,"tokens_out":1115,"duration_ms":8947,"temperature":1.0,"reasoning_tokens":1026,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:33:46.296852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all gem-free graphs on 23 edges by a backtracking generator that rejects any graph containing $H_5$, and compute their spectral radii. The theorem is false if any graph other than $S_{13,2}$ and $S^2_{14,2}$ has spectral radius greater than $\\rho(S^2_{14,2})$. Equivalently, inspect the intermediate graphs $G'$, $G^{\\star}$, and $G_5$ constructed in Lemma 3.4 and Claims 3.1–3.2: if any of them contains $H_5$ on five vertices, the contradiction argument identifying $S^2_{14,2}$ collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.1, the Perron-vector edge-shift inequality: moving an edge from a vertex to one with a larger Perron coordinate strictly raises the spectral radius. Every structural contradiction in the proof is built on this move."},{"cited_title":"Zhang, L.G","cited_arxiv_id":null,"evidence_quote":"Independently proves Theorem 1.1 for the gem, establishing the extremal graph $S_{\\frac{m+3}{2},2}$ that the present runner-up result excludes."}],"review_version":1}