{"id":"df382bf4-bdf3-4658-8ab9-8ee3a1bab082","arxiv_id":"1908.01433","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A spectral inequality of Hoffman type is proved for weighted k-partite uniform r-hypergraphs when r is even and p ≥ r.","lead":"This short paper extends Hoffman's classic bound on graph colorings to weighted uniform hypergraphs with an even number of vertices per edge. The proof yields a spectral ratio inequality for k-partite hypergraphs and a counterexample showing such a bound cannot depend only on the chromatic number.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the main inequality is proved correctly modulo a harmless typo in the range of the permutation set.","rationale":"The reader's verdict is ACCEPT with high confidence, and I concur. The proof of Theorem 1 is a clean averaging argument: inequality (2) is valid for every vector, summing over permutations yields the exact identity (5), and the Power Mean inequality (6) gives the needed upper bound because p ≥ r. The subsequent algebra in (7) correctly converts this into the claimed ratio inequality after dividing by the negative minima. The only gap in exposition is that Eq (8) is written as an upper bound, but equality follows by the constant vector, so the proof is easily completed. The typo in the definition of P (permutations of [r] rather than [k]) is unambiguous from the context and does not affect the mathematics. The p < r case being open is a genuine boundary of the theorem, not a flaw, since the theorem explicitly assumes p ≥ r. The 2-chromatic example in Section 2 is consistent and supports the necessity of the k-partite hypothesis. Therefore no change to the reader's verdict is warranted.","tokens_in":6499,"tokens_out":24304,"duration_ms":270758,"concrete_test":"Recompute Eq (5) with σ ranging over permutations of [k] for a small case, e.g., r = 4, k = 5, with G a single edge, to confirm the factor (k-r)! and the sign. Independently verify Eq (8) by substituting u_j = k^{-1/p} into P_{K^k_r}(u) and checking it attains the displayed upper bound, which supplies the omitted lower bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I do not find a load-bearing concern. The central averaging argument proving Theorem 1 is valid under its stated hypotheses p ≥ r and even r. The Power Mean step in (6) runs in the correct direction because r/p ≤ 1, and the sign bookkeeping in (7) checks out when dividing by the two negative minima. Equation (8) states only an upper bound, but the omitted lower bound is immediate from the constant vector u_j = k^{-1/p}, which gives equality in both Maclaurin's inequality and the Power Mean inequality. The permutation set is misprinted as [r] instead of [k], but every subsequent count (k!, (k-r)!, and the sum over σ of y_{σ(η(i))}) shows the intended set, so this is a cosmetic typo. The p < r regime is explicitly left open in Problem 4, so it does not undermine the theorem as stated. Section 2's example correctly shows why replacing k-partite by k-chromatic is not possible. No circularity, missing support, or overclaim was found.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Hoffman's classical chromatic-number eigenvalue bound to weighted uniform r-graphs. For an even integer r, real p ≥ r, integer k ≥ r, and a weighted k-partite r-graph G^w, Theorem 1 asserts that λ^{(p)}(G^w)/λ_min^{(p)}(G^w) ≥ λ^{(p)}(K^k_r)/λ_min^{(p)}(K^k_r), with equality for regular complete k-partite r-graphs. The proof averages the polynomial form P_G over the k! permutations of the partite classes and combines the Power Mean inequality with Maclaurin's inequality; the sign handling uses that λ_min is negative for even r. Section 2 constructs a family of 2-chromatic 4-graphs for which λ^{(p)}/|λ_min^{(p)}| grows linearly in the order, showing that replacing 'k-partite' by 'k-chromatic' is impossible. Two open problems are posed: meaningful extensions for odd r and for 1 ≤ p < r.","tokens_in":6685,"tokens_out":13187,"duration_ms":126540,"significance":"The main result is a genuine and natural generalization of Hoffman's bound to hypergraphs, obtained by a short and transparent argument that relies only on standard inequalities. The paper is careful to state the range of p and to flag the unknown value of λ_min(K^k_r); the counterexample in Section 2 is a useful negative result. The proof of the inequality is essentially complete, and the equality claim is supported except for a routine omitted calculation. These strengths make the paper a solid contribution to spectral hypergraph theory, suitable for a combinatorics or linear-algebra journal.","major_comments":[],"minor_comments":[{"comment":"In the proof of Theorem 1, P is defined as the set of all permutations σ:[r]→[r]; however, the subsequent counting of permutations, such as the r!(k-r)! permutations mapping {η(i1),...,η(ir)} onto {j1,...,jr}, is only valid for permutations of [k]. Please correct the definition to σ:[k]→[k] (or equivalently state the intended set explicitly), since as written the proof is not internally consistent.","section":"Proof of Theorem 1, definition of P"},{"comment":"The proof of equation (10) is omitted with the note that it 'goes along the same lines.' Since equality in (3) for complete regular k-partite r-graphs is part of the theorem statement, please include the argument or at least give the explicit analogue of the argument for λ_min, indicating that the same choice of x and the same two-sided bounding prove the equality for λ^{(p)} as well.","section":"Proof of Theorem 1, equation (10)"},{"comment":"The statement λ_min^{(p)}(G) = Ω(n^{3-4/p}) is not meaningful as written because λ_min^{(p)}(G) is negative, while the Ω-notation normally refers to positive quantities. The intended statement is |λ_min^{(p)}(G)| = Ω(n^{3-4/p}) (or λ_min^{(p)}(G) = -Ω(n^{3-4/p})). Please correct this statement and the analogous line in the proof.","section":"Proposition 2, equation (13)"},{"comment":"The citation 'the Rayleigh-Ritz theorem (see [3], Theorem 4.2.4)' appears to be a misreference; reference [3] is Hoffman's 1970 paper, whereas the Rayleigh-Ritz theorem is in [4] (Horn and Johnson, Matrix Analysis). Please correct the citation.","section":"Introduction, Rayleigh-Ritz citation"},{"comment":"The superscript (p) is used inconsistently: for example, equation (8) writes λ(K^k_r) without the superscript, and the paragraph after Theorem 1 refers to λ(K^k_r) without clarifying that the p-dependent eigenvalue is meant. Please standardize the notation to avoid confusion.","section":"Various locations, notation"},{"comment":"In the proof of Proposition 2, strict inequalities are used where weak inequalities would be more precise, e.g., '2S2(A) > ...' and '2S2(B) < ...'. Since the preceding estimates are derived from Maclaurin's and Power Mean inequalities, which are non-strict, please replace the strict signs with ≥ and ≤ as appropriate.","section":"Proof of Proposition 2"}],"recommendation":"minor_revision","confidential_remarks":"The paper's central claim is sound; the issues are local typos and notation problems. I recommend minor revision. The proof can be made fully rigorous with small corrections. No concerns about novelty or attribution; the methods and references are appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this short note with some care, and I think the reader's report is about right. The central result—Theorem 1—generalizes Hoffman's bound to weighted even-uniform r-graphs, and the proof is genuinely clean. The averaging argument is a nice extension of the author's earlier eigenvector method, and the weighted setting plus the equality case for regular complete k-partite r-graphs are real additions. The infinite family of 2-chromatic 4-graphs in Section 2 is a good counterexample: it shows the k-partite hypothesis cannot simply be replaced by k-chromatic, because the ratio λ/|λ_min| grows unboundedly. I checked the main steps of the proof, and they hold. The Power Mean inequality is used in the right direction exactly because p≥r, and the sign bookkeeping works because both λ_min values are negative. The omitted proof of equation (10) is not a problem—it is genuinely the same lines as (9), and the statement is true.\n\nThe soft spots are minor and mostly cosmetic. The permutation set P is misprinted as [r]→[r] when it is used as permutations of [k]; the subsequent counting (k!, (k-r)!, the sum over σ) makes the intended set clear, so this is harmless. Proposition 2 writes λ_min = Ω(n^{3-4/p}) for a negative quantity, which is informal—really one should say |λ_min| = Ω(...) or use the sign explicitly. The dangling 'Proposition ??' is a typographical artifact. The one substantive caveat is that the right side of (3) depends on λ_min(p)(K^k_r), which is not known exactly, although the author says a forthcoming paper will determine its order of magnitude. That limits immediate numerical applications, but it doesn't affect the validity of the theorem.\n\nThe p<r regime is left open in Problem 4, and the author does not claim to cover it. That is honest. I also see no circularity: the proof uses standard inequalities and prior definitions from [9] and [10], not the theorem itself. The citation pattern is appropriate.\n\nWho this is for: anyone working on spectral hypergraph theory or spectral bounds for chromatic/partition properties. It is not a big paper, but it is a solid, correct extension of a classical result. I would send it to a referee rather than desk-reject it; with minor revisions (fix the typo, include or spell out (10), fix the Ω notation) it should be publishable as is.","headline":"A clean, correct generalization of Hoffman's bound to weighted even-uniform hypergraphs; the p≥r caveat is real but honestly flagged, and the proof holds up.","tokens_in":7194,"tokens_out":3033,"would_cite":true,"duration_ms":30761,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A42","05C65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Hoffman's eigenvalue bound for graph colorings carries over to weighted even-rank hypergraphs: for every weighted $k$-partite $r$-graph with even $r$ and $p \\ge r$, the spectral ratio $\\lambda^{(p)}/\\lambda^{(p)}_{\\min}$ is at least the…","keywords":["Hoffman's bound","hypergraph eigenvalues","k-partite hypergraphs","weighted hypergraphs","ℓ^p-spectral radius","chromatic number","complete k-partite r-graphs","polynomial form"],"falsifier":"Take the complete 4-partite 4-graph on five singleton parts (five edges, one per 4-subset of parts), change one edge weight from 1 to 2, and compute both sides of inequality (3) at $p = 2$ by numerically optimizing the polynomial form $P_{G^w}$ on the $\\ell^2$ unit sphere. If the perturbed ratio comes out strictly smaller than the complete ratio (larger in magnitude), the claimed bound fails below the $p \\ge r$ threshold and Problem 4 is answered negatively; if it holds for every single-edge perturbation, the restriction is likely technical, and the same test at $p = 4$ would confirm the theorem on its proven range.","tokens_in":6303,"feed_emoji":"📏","tokens_out":37846,"duration_ms":319810,"temperature":0.7,"pith_summary":"Hoffman's inequality for graphs says the chromatic number satisfies $\\chi(G) \\ge 1 - \\lambda(G)/\\lambda_{\\min}(G)$, tying colorability to the two extreme adjacency eigenvalues. This note proves the analogous statement for weighted $r$-uniform hypergraphs with even $r$: if $G^w$ is $k$-partite (vertices split into $k$ classes, every edge meeting each class at most once), then for every $p \\ge r$, $\\lambda^{(p)}(G^w)/\\lambda^{(p)}_{\\min}(G^w) \\ge \\lambda^{(p)}(K^k_r)/\\lambda^{(p)}_{\\min}(K^k_r)$, where the two quantities are the maximum and minimum of the hypergraph's polynomial form on the unit $\\ell^p$ sphere, and $K^k_r$ is the complete $k$-partite $r$-graph; regular complete $k$-partite $r$-graphs attain equality. Setting $r = p = 2$ with unit weights recovers Hoffman's inequality exactly. The result matters because the bound is sharp on a natural class and because the paper shows the hypothesis $k$-partite cannot be relaxed to $k$-chromatic: explicit 2-chromatic 4-graphs have spectral ratio growing without bound.","feed_headline":"Hoffman's bound holds for weighted even-rank hypergraphs","feed_subtitle":"In k-partite weighted r-graphs the eigenvalue ratio is minimized by the complete case; r=p=2 gives Hoffman's bound.","key_machinery":"The load-bearing object is the polynomial form $P_{G^w}(x) = r! \\sum_{\\{i_1,\\dots,i_r\\} \\in E(G)} w_{i_1\\dots i_r} x_{i_1}\\cdots x_{i_r}$ on $\\mathbb{R}^n$, whose extrema on the unit $\\ell^p$ sphere define $\\lambda^{(p)}(G^w)$ and $\\lambda^{(p)}_{\\min}(G^w)$. The proof's engine is a permutation collapse: write $\\eta(i)$ for the part of vertex $i$, let $x$ maximize $P_{G^w}$ and $y$ minimize the complete $k$-partite form, and form $z_{\\sigma,i} = x_i y_{\\sigma(\\eta(i))}$ for every permutation $\\sigma$ of the $k$ parts; summing the pointwise bound $P_{G^w}(z_\\sigma) \\ge \\lambda^{(p)}_{\\min}(G^w)|z_\\sigma|_p^r$ over all $k!$ permutations turns the cross terms into $(k-r)!\\,\\lambda^{(p)}_{\\min}(K^k_r)\\,\\lambda^{(p)}(G^w)$. The Power Mean inequality — which needs $r/p \\le 1$, hence $p \\ge r$ — then caps the accumulated $\\ell^p$ sums at $k^{1-r/p}(k-1)!$, and Maclaurin's inequality yields $\\lambda^{(p)}(K^k_r) = k^{1-r/p}(k-1)\\cdots(k-r+1)$; assembling these pieces gives the ratio bound.","core_discovery":"The paper's central claim is Theorem 1: for even $r \\ge 2$, integers $k \\ge r$, real $p \\ge r$, and any weighted $k$-partite $r$-graph $G^w$, the ratio inequality $\\lambda^{(p)}(G^w)/\\lambda^{(p)}_{\\min}(G^w) \\ge \\lambda^{(p)}(K^k_r)/\\lambda^{(p)}_{\\min}(K^k_r)$ holds, with equality for regular complete $k$-partite $r$-graphs. Since $\\lambda^{(p)}_{\\min}$ is negative, the inequality bounds the magnitude of the extreme-eigenvalue ratio from above by the complete $k$-partite value; for ordinary graphs at $r = p = 2$ it reads $1 - \\lambda/\\lambda_{\\min} \\le k$, which with $k = \\chi(G)$ is Hoffman's inequality. The weighted form also extends a matrix inequality of Lovász. The proof pins down half of the sharp constant, $\\lambda^{(p)}(K^k_r) = k^{1-r/p}(k-1)\\cdots(k-r+1)$, while the corresponding minimum $\\lambda^{(p)}_{\\min}(K^k_r)$ remains an explicitly identified unknown whose order of magnitude is announced for a sequel.","pith_inferences":["A decisive test of the $p \\ge r$ restriction is the unweighted complete $k$-partite $r$-graph itself: since the paper gives $\\lambda^{(p)}(K^k_r)$ exactly but leaves $\\lambda^{(p)}_{\\min}(K^k_r)$ open, computing that minimum for $1 \\le p < r$ would likely settle Problem 4 and reveal whether the reversed Power-Mean direction is a real obstruction or a proof artifact.","Read against the graph case, the 2-chromatic example marks a genuine graph/hypergraph divide: bipartite graphs have perfectly symmetric spectra (ratio exactly $-1$), whereas 2-chromatic 4-graphs can have unbounded ratio — so for $r \\ge 4$ the structural parameter that spectral methods see is k-partiteness, not k-colorability.","A quantitative stability statement suggests itself: for graphs, Hoffman-type bounds are tight on many graphs, and the paper's equality case suggests that weighted $k$-partite $r$-graphs whose ratio comes close to the complete $k$-partite value should be close to a regular complete $k$-partite $r$-graph in structure (balanced parts, nearly constant weights) — a near-equality version of the paper's "],"forward_implications":["At $r = p = 2$ with unit weights the theorem is Hoffman's inequality: it gives $1 - \\lambda(G)/\\lambda_{\\min}(G) \\le k$ for every $k$-partite graph, and taking $k = \\chi(G)$ returns $\\chi(G) \\ge 1 - \\lambda(G)/\\lambda_{\\min}(G)$; allowing edge weights extends a matrix version of the bound due to Lovász.","For any weighted $k$-partite $r$-graph with even $r$ and $p \\ge r$, the magnitude of the spectral ratio is at most the complete $k$-partite value, $|\\lambda^{(p)}(G^w)/\\lambda^{(p)}_{\\min}(G^w)| \\le \\lambda^{(p)}(K^k_r)/|\\lambda^{(p)}_{\\min}(K^k_r)|$, and regular complete $k$-partite $r$-graphs attain equality, so that value is the sharp constant.","The hypothesis 'k-partite' is essential: the 4-graphs on $2n$ vertices whose edges are the 4-sets containing exactly two vertices from each of two equal classes are 2-chromatic yet satisfy $\\lambda^{(p)}(G)/|\\lambda^{(p)}_{\\min}(G)| = \\Omega(n)$, so no function of the chromatic number alone can bound the spectral ratio for $r \\ge 4$.","For odd $r$ the theorem holds only trivially, since $\\lambda^{(p)}_{\\min}(G) = -\\lambda^{(p)}(G)$ forces the ratio to be $-1$; the paper's open problems ask for a meaningful odd-$r$ extension and for the even-$r$ regime $1 \\le p < r$.","The proof fixes the numerator of the sharp constant, $\\lambda^{(p)}(K^k_r) = k^{1-r/p}(k-1)\\cdots(k-r+1)$, while the denominator $\\lambda^{(p)}_{\\min}(K^k_r)$ remains an identified unknown whose order of magnitude the paper announces for a forthcoming paper."],"supporting_citations":[{"why":"Hoffman's original inequality $\\chi(G) \\ge 1 - \\lambda(G)/\\lambda_{\\min}(G)$ — the statement Theorem 1 generalizes and the $r = p = 2$ case recovers.","marker":"[3]"},{"why":"Lovász's matrix version of the pointwise inequality (2), which the introduction of edge weights extends.","marker":"[8]"},{"why":"Contributes the central proof device — summing the pointwise inequality over permutations of the part labels — on which the proof of Theorem 1 is based.","marker":"[9]"},{"why":"Earlier use of a similar permutation idea for hypergraph coloring, the closest prior spectral-coloring result for $r$-graphs.","marker":"[7]"},{"why":"Introduces the parameter $\\lambda^{(p)}(G)$ (with Keevash, Lenz, and Mubayi) that the theorem bounds.","marker":"[5]"},{"why":"Introduces $\\lambda^{(p)}_{\\min}(G)$ and the analytic framework for uniform hypergraphs used throughout.","marker":"[10]"},{"why":"Lemma 9.6.2 documents the tightness and equality cases of Hoffman's bound that the paper's equality statement mirrors.","marker":"[2]"}],"fun_headline_variants":["Hoffman bound proven for even-rank weighted hypergraphs","Weighted even-uniform hypergraphs obey Hoffman's inequality","Spectral ratio bound for even-rank weighted hypergraphs","Hoffman's inequality extended to even-rank hypergraphs","Eigenvalue bound for weighted r-graphs with even r"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $p \\ge r$ and $r$ is even: the Power Mean step in equation (6) needs $r/p \\le 1$ to point the right way, the case $1 \\le p < r$ is left open as Problem 4, and for odd $r$ the statement holds only in the trivial sense that $\\lambda^{(p)}_{\\min}(G) = -\\lambda^{(p)}(G)$.","fun_headline_variants_meta":{"raw":{"variants":["Hoffman bound proven for even-rank weighted hypergraphs","Weighted even-uniform hypergraphs obey Hoffman's inequality","Spectral ratio bound for even-rank weighted hypergraphs","Hoffman's inequality extended to even-rank hypergraphs","Eigenvalue bound for weighted r-graphs with even r"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1222,"prompt_tokens":905,"completion_tokens":317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":236}},"tokens_in":521,"tokens_out":317,"duration_ms":4153,"temperature":1.0,"reasoning_tokens":236,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:13:29.537134+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the complete 4-partite 4-graph on five singleton parts (five edges, one per 4-subset of parts), change one edge weight from 1 to 2, and compute both sides of inequality (3) at $p = 2$ by numerically optimizing the polynomial form $P_{G^w}$ on the $\\ell^2$ unit sphere. If the perturbed ratio comes out strictly smaller than the complete ratio (larger in magnitude), the claimed bound fails below the $p \\ge r$ threshold and Problem 4 is answered negatively; if it holds for every single-edge perturbation, the restriction is likely technical, and the same test at $p = 4$ would confirm the theorem on its proven range.","supporting_citations":[{"cited_title":"Hoffman, On eigenvalues and colorings of graphs, in Graph Theory and its Applica- tions, Academic Press, New York (1970), pp","cited_arxiv_id":null,"evidence_quote":"Hoffman's original inequality $\\chi(G) \\ge 1 - \\lambda(G)/\\lambda_{\\min}(G)$ — the statement Theorem 1 generalizes and the $r = p = 2$ case recovers."},{"cited_title":"Lovász, On the Shannon capacity of a graph","cited_arxiv_id":null,"evidence_quote":"Lovász's matrix version of the pointwise inequality (2), which the introduction of edge weights extends."},{"cited_title":"Nikiforov , Chromatic number and spectral radius, Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Contributes the central proof device — summing the pointwise inequality over permutations of the part labels — on which the proof of Theorem 1 is based."},{"cited_title":"Kenter, Necessary spectral conditions for coloring h ypergraphs, J","cited_arxiv_id":null,"evidence_quote":"Earlier use of a similar permutation idea for hypergraph coloring, the closest prior spectral-coloring result for $r$-graphs."},{"cited_title":"Keevash, J","cited_arxiv_id":null,"evidence_quote":"Introduces the parameter $\\lambda^{(p)}(G)$ (with Keevash, Lenz, and Mubayi) that the theorem bounds."},{"cited_title":"Nikiforov , Analytic methods for uniform hypergraphs, Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Introduces $\\lambda^{(p)}_{\\min}(G)$ and the analytic framework for uniform hypergraphs used throughout."},{"cited_title":"Godsil and G","cited_arxiv_id":null,"evidence_quote":"Lemma 9.6.2 documents the tightness and equality cases of Hoffman's bound that the paper's equality statement mirrors."}],"review_version":1}