{"id":"a4171d8f-bda0-4e13-9b6e-5c371941bd7f","arxiv_id":"2411.18120","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two discrete rectangular tori with equal Laplacian spectra are necessarily isomorphic.","lead":"This paper proves that two discrete rectangular tori (grid-shaped graphs with wraparound) with the same Laplacian spectrum are always isomorphic, the discrete analogue of a classical continuous result. The proof is short and constructive: use the smallest nonzero eigenvalue to identify the largest cycle factor, then peel factors off one by one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 2 depends on an unstated convention for C_2; under the default simple-graph reading the peeling argument collapses.","rationale":"The paper's central contribution is plausible and the theta-function machinery is sound: Lemma 2 follows from the standard product formula, and the cancellation step is legitimate because Theta is strictly positive. The weakest point is exactly the one the reader identifies: the spectrum formula for C_m at m = 2 and the use of algebraic connectivity to identify the largest factor. Under the simple-graph convention, C_2 = K_2 has Laplacian spectrum {0,2}, so a(K_2) = a(C_4) = 2; consequently the equality a(T) = a(T~) does not imply equality of the largest indices when one torus has a K_2 factor and the other has a C_4 factor. Furthermore C_4 is isomorphic to K_2 x K_2, so Lemma 3's unique representation is false in the simple convention and Proposition 1 itself fails for the pair C_4 and C_2 x C_2, which are the same graph. The theorem 'isospectral iff isomorphic' may still be true because the problematic pairs are isomorphic and the prime factorization theorem underlies the graph class, but the current proof does not establish it without an explicit convention. A secondary gap is that Proposition 1's final inequality needs the algebraic-connectivity equality of the remaining pieces to identify the largest remaining factor; this is not written out but is recoverable under the multigraph convention. I therefore concur with the reader's CONDITIONAL verdict and see no reason to strengthen the objection.","tokens_in":6815,"tokens_out":24913,"duration_ms":227460,"concrete_test":"In the simple-graph convention, compare T = C_4 and T~ = C_2 x C_2: both are the same four-cycle, so they have identical Laplacian polynomial, yet Proposition 1 asserts p = p~, which fails (1 versus 2). This one computation settles that the proof as written is invalid for simple graphs; the paper must either define C_2 as the two-edge multigraph (and justify the multigraph factorization) or restrict all factors to m_i >= 3 and treat C_4 separately.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main proof peels off the largest factor by comparing the algebraic connectivity 4 sin^2(pi/m_p). Section 2.5 states that C_m has spectrum {4 sin^2(pi j/m)}, j = 0, ..., m-1; for m = 2 this formula gives {0,4}, the Laplacian spectrum of the two-vertex multigraph with two parallel edges, not of the simple graph K_2, whose spectrum is {0,2}. The paper defines graphs only as undirected and lets a_uv count edges, but never states whether C_2 is this multigraph or K_2. Under the default simple-graph reading, a(C_2) = a(C_4) = 2, so the first step of Proposition 1, namely 4 sin^2(pi/m_p) = 4 sin^2(pi/m~_p) hence m_p = m~_p, fails. Moreover Lemma 3's uniqueness statement fails because C_4 is isomorphic to K_2 x K_2. Concretely, C_4 and C_2 x C_2 are the same simple graph, so they are isospectral with dimensions 1 and 2, contradicting Proposition 1. The theorem may still be true under either convention, but the proof as written is sound only if C_2 is explicitly declared to be the two-edge multigraph and multigraph versions of Sabidussi-Vizing and the algebraic connectivity properties are justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that two discrete rectangular tori—defined as Cartesian products of cyclic graphs C_{m_1} × ⋯ × C_{m_p} with 2 ≤ m_1 ≤ ⋯ ≤ m_p—are isospectral if and only if they are isomorphic. The proof introduces a theta function Θ_G(t) that encodes the Laplacian spectrum, uses the multiplicative property of Θ under Cartesian products, and peels off the largest cycle factor by identifying the algebraic connectivity 4 sin²(π/m_p). A uniqueness result for the factorization of tori (Lemma 3) is used to conclude the proof by induction. The paper also claims that the dimension of a torus is an audible (spectral) invariant.","tokens_in":7063,"tokens_out":8703,"duration_ms":74625,"significance":"If the proof is correct, the result provides a clean discrete analogue of the classical theorem that isospectral flat tori are isometric, and it adds to the small family of graph classes that are determined by their Laplacian spectrum. The argument is self-contained, short, and does not rely on fitted parameters or numerical computation. Its main ingredients—the Laplacian spectrum of Cartesian products, Sabidussi–Vizing factorization, and a theta-function recovery of the spectrum—are standard and correctly cited. The result could be useful for the Buser problem on genus-two Riemann surfaces via the discrete theta-graph analogues mentioned in the introduction, making the paper of interest to the spectral graph theory community.","major_comments":[{"comment":"The definition of C_2 is ambiguous and the paper's proof depends critically on a convention that is never stated. The spectrum formula {4 sin²(πj/m) : j = 0, …, m−1} gives {0, 4} for m = 2, which is the Laplacian spectrum of the two-vertex multigraph with two parallel edges, not of the simple graph K_2, whose spectrum is {0, 2}. Under the standard simple-graph reading, C_2 = K_2 and C_4 ≅ K_2 × K_2 are the same graph, so the representation of a torus as C_{m_1} × ⋯ × C_{m_p} is not unique (C_4 has both p = 1 and p = 2 representations), and Proposition 1's claim that dimension is a spectral invariant is false. Moreover, the algebraic connectivity comparison 4 sin²(π/m_p) = 4 sin²(π/m̃_p̃) fails to imply m_p = m̃_p̃ when the values coincide, as they do for m = 2 and m = 4 under the simple-graph convention. The proof of Theorem 2 relies on this equality to peel off factors. The authors must either explicitly define C_2 as the two-vertex multigraph with two parallel edges and verify that the cited Sabidussi–Vizing theorem and the spectrum formula apply to the resulting class of multigraph Cartesian products, or they must modify the class of tori to exclude C_2 and give a proof that handles C_4 without relying on a one-to-one map between m and a(C_m).","section":"§2.4–§2.5, Proposition 1, Theorem 2"},{"comment":"The uniqueness proof for the representation T = C_{m_1} × ⋯ × C_{m_p} is not rigorous as written. The paper invokes the Sabidussi–Vizing theorem, which is normally stated for simple graphs, while Section 2.1 allows a_{uv} to count multiple edges. The argument that C_m is prime for m ≠ 4 and that C_4 can be replaced by K_2 × K_2 assumes the simple-graph setting, but then the cyclic factor C_4 is not prime because C_4 = K_2 × K_2. If C_2 is instead taken to be a multigraph, then K_2 is not the same as C_2, and the replacement step needs a justification for why the multigraph version of Sabidussi–Vizing still holds and why the representation is unique among cyclic factors. The lemma is load-bearing because Theorem 2's induction step requires a well-defined and unique factorization; the paper should state the exact class of graphs under consideration and prove or cite a unique factorization theorem for that class.","section":"Lemma 3"}],"minor_comments":[{"comment":"The affiliation contains a typo: \"Unoversity\" should be \"University\".","section":"Title page"},{"comment":"The sentence \"In these papers, In these papers, it was shown...\" contains a duplicated phrase that should be removed.","section":"Introduction"},{"comment":"The grammar \"Let G1 and G2 are two finite graphs\" should be \"Let G1 and G2 be two finite graphs\".","section":"§2.3, Lemma 2"},{"comment":"The phrase \"If p = p̃, we have what needs to be done\" is awkward; it should read \"If p = p̃, there is nothing to prove\".","section":"§2.6, Proposition 1"},{"comment":"Reference [12] cites \"Charter 6\" instead of \"Chapter 6\".","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The single substantive obstacle is the C_2 convention. The authors can likely fix this by explicitly adopting the multigraph convention for C_2 and invoking (or proving) the necessary unique factorization theorem for multigraph Cartesian products, or by excluding C_2 and reworking the proof for the remaining cases. The rest of the derivation is sound and the result is likely correct under either adequate convention, so I recommend major revision rather than rejection. I did not find any evidence of circularity or fabricated reasoning; the proof is a straightforward derivation from standard theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper proves that two discrete rectangular tori (Cartesian products of cycles) are isospectral iff they are isomorphic, extending the known continuous flat-torus result to the graph setting. The proof is elegant: the theta function turns isospectrality into equality of exponential sums, the algebraic connectivity peels off the largest cycle factor, and induction finishes. That part is genuinely nice, and the theorem is new.\n\nThe problem is the treatment of C_2. The paper allows m_i = 2 and uses the spectrum formula 4 sin^2(pi j/m_i), which for m=2 gives {0,4}. That is the spectrum of the two-vertex multigraph with two parallel edges, not of the simple graph K_2, whose Laplacian spectrum is {0,2}. The paper never states which convention is intended. Under the standard simple-graph reading, the proof collapses immediately: K_2 × K_2 is isomorphic to C_4, so the torus C_4 has two valid representations, one with p=1 and one with p=2. Proposition 1, which claims isospectral tori have the same dimension, is then false, and the step where mp = m~p from equality of algebraic connectivities fails because a(C_2) = a(C_4) = 2. Lemma 3's uniqueness of representation is also false in this reading.\n\nThe reader's stress-test note is right, and it lands on reading the paper. The fix is local. If the authors restrict to m_i >= 3, the spectrum formula is correct, the algebraic connectivity is strictly decreasing in m, and the induction goes through. Alternatively, they can explicitly define C_2 as a two-edge multigraph and then justify the Sabidussi-Vizing and square-lemma arguments in that category. Either way, the gap is a definitional oversight, not a dead end. The central derivation with theta functions and the peeling argument is sound once the graph class is pinned down.\n\nFor a reader in spectral graph theory, this is a worthwhile paper. It deserves a serious referee: the theorem is plausible, the method is transparent, and the needed revisions are clearly scoped. I would not desk-reject it. I'd send it out with a request to fix the C_2/C_4 convention and re-check Proposition 1 and Lemma 3 under the stated definitions.","headline":"A clean and likely true theorem for discrete tori, but the proof as written has an unstated C_2/C_4 convention problem that breaks the peeling argument until fixed.","tokens_in":7590,"tokens_out":6766,"would_cite":true,"duration_ms":56736,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C76"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Laplacian spectrum of a discrete rectangular torus determines the torus completely: two such tori are isospectral if and only if they are isomorphic.","keywords":["Laplacian spectrum","isospectral graphs","discrete torus","Cartesian product","spectral determination","theta function","algebraic connectivity"],"falsifier":"Enumerate all ordered tuples $(m_1,\\dots,m_p)$ with small entries and compare the multisets $M=\\{\\sum_{i=1}^p 4\\sin^2(\\pi j_i/m_i): 0\\le j_i<m_i\\}$; any two distinct tuples with the same multiset would falsify Theorem 2. A cheap first check is the spectrum of $K_2\\times C_3$, whose true eigenvalues should be compared with the formula implied by the proof when $m=2$.","tokens_in":6603,"feed_emoji":"👂","tokens_out":8909,"duration_ms":76805,"temperature":0.7,"pith_summary":"The paper proves that the Laplacian spectrum of a discrete rectangular torus—a graph formed as the Cartesian product of cycles $C_{m_1}\\times\\cdots\\times C_{m_p}$—determines the torus up to isomorphism: two such graphs are isospectral if and only if they are isomorphic. This answers the discrete drum-typing question for the simplest multidimensional graph tori, saying that from the eigenvalues one can recover the dimension and the full list of cycle lengths. The result is in the same spirit as the known continuous theorem for rectangular flat tori, and the proof is short because two spectral invariants do all the work: the smallest positive Laplacian eigenvalue identifies the longest cycle factor, and the $\\theta$ function splits the product into factors.","feed_headline":"Spectrum alone determines a discrete rectangular torus","feed_subtitle":"Cycle lengths are audible in the eigenvalues: matching spectra force matching tori.","key_machinery":"The proof is carried by the $\\theta$ function $\\Theta_G(t)=\\sum_{\\lambda\\in S_G}e^{-\\lambda t}$, which encodes the full Laplacian spectrum and satisfies $\\Theta_{G_1\\times G_2}(t)=\\Theta_{G_1}(t)\\Theta_{G_2}(t)$ for Cartesian products. Together with the algebraic connectivity $a(G)$, the smallest positive Laplacian eigenvalue, this gives a peeling argument: for a torus $T=C_{m_1}\\times\\cdots\\times C_{m_p}$ with $m_1\\le\\cdots\\le m_p$, the value $a(T)=4\\sin^2(\\pi/m_p)$ determines the largest cycle length $m_p$, and cancellation of the common $\\theta$ factor $\\Theta_{C_{m_p}}(t)$ reduces spectral equality of tori to spectral equality of the remaining products.","core_discovery":"The paper's central claim is Theorem 2: two discrete rectangular tori are isospectral if and only if they are isomorphic. A discrete rectangular torus is the Cartesian product $C_{m_1}\\times\\cdots\\times C_{m_p}$ with each $m_i\\ge 2$, and the theorem says the Laplacian spectrum knows the ordered tuple $(m_1,\\dots,m_p)$. The proof first shows the dimension is audible (Proposition 1), then peels factors one at a time: the algebraic connectivity $4\\sin^2(\\pi/m_p)$ reveals the largest cycle length, the $\\theta$-function identity $\\Theta_{G\\times H}=\\Theta_G\\Theta_H$ lets that factor be cancelled, and induction identifies all remaining factors. The conclusion also records that the result does not extend to all circulant graphs, since isospectral non-isomorphic circulant graphs exist on 20 vertices.","pith_inferences":["The peeling argument is more general than the torus setting: spectral equality of two Cartesian products with a common factor forces equality of the complementary factors, since the theta function lets the common factor be cancelled.","The paper leaves open the analogue for discrete tori built from arbitrary parallelepiped lattices; a testable next step is whether the algebraic connectivity still identifies the largest side in that class.","The existence of isospectral non-isomorphic circulant graphs on 20 vertices puts the torus result near a sharp boundary, since a torus is the special circulant graph coming from a product of cycles.","No exhaustive search for small tuples is given; running one would confirm the core identity or find a hidden counterexample."],"forward_implications":["Any two isospectral discrete rectangular tori have the same dimension and the same ordered tuple of cycle lengths.","The proof gives a recursive way to read the tuple $(m_1,\\dots,m_p)$ from the Laplacian spectrum, so the torus can be reconstructed up to isomorphism from its eigenvalues.","Every discrete rectangular torus is determined by its Laplacian spectrum, adding a large family to the known examples of graphs that are spectrally determined.","Any graph invariant that depends only on the ordered tuple $(m_1,\\dots,m_p)$, such as the number of vertices, is automatically a spectral invariant of the torus."],"supporting_citations":[{"why":"Supplies the spectrum of a Cartesian product as all sums of factor eigenvalues, and the Laplacian eigenvalues of cycle graphs.","marker":"[22]"},{"why":"Provides the uniqueness theorem for prime factorization of connected graphs under the Cartesian product, used to make the torus representation unique.","marker":"[12]"},{"why":"Establishes uniqueness of the Cartesian prime factorization, the basis of Lemma 3.","marker":"[24]"},{"why":"Independent proof of the same uniqueness theorem, also cited for Lemma 3.","marker":"[27]"},{"why":"Defines algebraic connectivity and gives the product formula $a(G_1\\times G_2)=\\min(a(G_1),a(G_2))$, used to identify the largest cycle factor.","marker":"[9]"},{"why":"Justifies that the theta function completely determines the spectrum, making $\\Theta_G$ an equivalent spectral invariant.","marker":"[3]"}],"fun_headline_variants":["Spectrum fingerprints discrete rectangular tori","Matching spectra force matching rectangular tori","Torus eigenvalues encode its side lengths uniquely","Rectangular tori are spectrally unique","Hear the torus's eigenvalues, name its shape"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every factor $C_m$ in the product has the standard Laplacian spectrum $\\{4\\sin^2(\\pi j/m)\\}$, including the edge case $m=2$ where that formula requires the double-edge convention; the paper never states this convention explicitly.","fun_headline_variants_meta":{"raw":{"variants":["Spectrum fingerprints discrete rectangular tori","Matching spectra force matching rectangular tori","Torus eigenvalues encode its side lengths uniquely","Rectangular tori are spectrally unique","Hear the torus's eigenvalues, name its shape"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001065,"raw_usage":{"total_tokens":4346,"prompt_tokens":708,"completion_tokens":3638,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":324,"completion_tokens_details":{"reasoning_tokens":3570}},"tokens_in":324,"tokens_out":3638,"duration_ms":24274,"temperature":1.0,"reasoning_tokens":3570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:30:59.551583+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all ordered tuples $(m_1,\\dots,m_p)$ with small entries and compare the multisets $M=\\{\\sum_{i=1}^p 4\\sin^2(\\pi j_i/m_i): 0\\le j_i<m_i\\}$; any two distinct tuples with the same multiset would falsify Theorem 2. A cheap first check is the spectrum of $K_2\\times C_3$, whose true eigenvalues should be compared with the formula implied by the proof when $m=2$.","supporting_citations":[{"cited_title":"Mohar, The Laplacian spectrum of graphs , in Graph theory, combinatorics, and applications 2, Ed","cited_arxiv_id":null,"evidence_quote":"Supplies the spectrum of a Cartesian product as all sums of factor eigenvalues, and the Laplacian eigenvalues of cycle graphs."},{"cited_title":"Imrich, S","cited_arxiv_id":null,"evidence_quote":"Provides the uniqueness theorem for prime factorization of connected graphs under the Cartesian product, used to make the torus representation unique."},{"cited_title":"Sabidussi, Graph multiplication, Math","cited_arxiv_id":null,"evidence_quote":"Establishes uniqueness of the Cartesian prime factorization, the basis of Lemma 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Independent proof of the same uniqueness theorem, also cited for Lemma 3."},{"cited_title":"Fiedler, Algebraic connectivity of graphs , Czech","cited_arxiv_id":null,"evidence_quote":"Defines algebraic connectivity and gives the product formula $a(G_1\\times G_2)=\\min(a(G_1),a(G_2))$, used to identify the largest cycle factor."},{"cited_title":"Brooks, Constructing isospectral manifolds , Amer","cited_arxiv_id":null,"evidence_quote":"Justifies that the theta function completely determines the spectrum, making $\\Theta_G$ an equivalent spectral invariant."}],"review_version":1}