One can hear a discrete rectangular torus

A.D. Mednykh; G.K. Sokolova; I.A. Mednykh

REVIEW 2 major objections 5 minor 28 references

One can hear a discrete rectangular torus

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.18120 v1 pith:DI6H73K6 submitted 2024-11-27 math.CO

A.D. Mednykh , I.A. Mednykh , G.K. Sokolova This is my paper · ORCID

classification math.CO MSC 05C5005C76

keywords LaplacianspectrumisospectralgraphsdiscretetorusCartesianproductspectraldeterminationthetafunctionalgebraicconnectivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $m=2$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

For 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[§2.4–§2.5, Proposition 1, Theorem 2] 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).
[Lemma 3] 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.

minor comments (5)

[Title page] The affiliation contains a typo: "Unoversity" should be "University".
[Introduction] The sentence "In these papers, In these papers, it was shown..." contains a duplicated phrase that should be removed.
[§2.3, Lemma 2] The grammar "Let G1 and G2 are two finite graphs" should be "Let G1 and G2 be two finite graphs".
[§2.6, Proposition 1] 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".
[References] Reference [12] cites "Charter 6" instead of "Chapter 6".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives the theorem from standard external results, with no fitted parameters or self-citation used as load-bearing evidence.

full rationale

The derivation chain is self-contained: Theorem 2 follows from the standard Laplacian spectrum formula for cycles, the known spectrum of Cartesian products (cited to Mohar [22]), and the Sabidussi-Vizing unique factorization theorem (cited to [12,24,27]). The theta-function arguments (Lemmas 1 and 2) are direct consequences of the spectrum and are not assumed to contain the target result. The peeling argument in Proposition 1 and Theorem 2 uses the algebraic connectivity and the number of vertices, both established spectral invariants, and then cancels common factors via the theta identity to reduce to lower-dimensional tori; this is a legitimate induction, not an equivalence to the conclusion. The only notable caveat is the convention for C_2 in Section 2.5, where the formula {4 sin^2(pi j/m)} differs from the simple graph K_2's Laplacian spectrum; however, this is a correctness/edge-case concern about an unstated convention, not a circularity. The self-citations [18] and [20] are motivational background and are not load-bearing for the proof. No fitted input is renamed as a prediction, and no result is justified solely by a self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof is a short derivation from standard theorems: the spectrum of a Cartesian product is the sumset of factor spectra (Mohar), the Sabidussi-Vizing unique factorization, and the fact that the theta function determines the spectrum. The only nonstandard input is the spectrum formula for C_m when m=2, which requires the multigraph interpretation.

assumptions (4)

standard math The Laplacian spectrum of a Cartesian product is the sumset of the spectra of the factors.
Invoked in Section 2.5 and the proof of Lemma 2, cited to Mohar [22].
domain assumption The cycle C_m has Laplacian spectrum {4 sin^2(π j/m)}.
Section 2.5; for m=2 this requires C_2 to be the 2-vertex multigraph with two parallel edges, which is not stated.
domain assumption The representation of a torus as a product of cycles is unique (Lemma 3).
Section 2.4; proven via Sabidussi-Vizing, but relies on the classification of prime cycles and the handling of C_4.
standard math The theta function Θ_G(t) determines the Laplacian spectrum.
Lemma 1, proved in Section 2.3 using linear independence of exponentials.

how reviews work

0 comments

Cite this review

Pith. "Pith review of One can hear a discrete rectangular torus." pith.science (2026). https://pith.science/paper/DI6H73K6

@misc{pith2026241118120,
  author       = {Pith},
  title        = {Pith review of: One can hear a discrete rectangular torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DI6H73K6}},
  note         = {Machine review of arXiv:2411.18120}
}

read the original abstract

In the present paper, we prove that two discrete rectangular tori are isospectral if and only if they are isomorphic.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Barden, H

D. Barden, H. Kang, Isospectral surfaces of genus two and three , Math. Proc. Camb. Phil. Soc., 153:1 (2012), 99 –110. MR2943668

work page 2012
[2]

Brooks, R

R. Brooks, R. Tse, Isospectral surfaces of small genus , Nagoya Math. J. 107 (1987), 13–24; MR0909246 Corrigendum: R. Brooks, R. Tse, Na goya Math. J. 117 (1990), 227. MR1044942

work page 1987
[3]

Brooks, Constructing isospectral manifolds , Amer

R. Brooks, Constructing isospectral manifolds , Amer. Math. Monthly, 95:9 (1988), 823–839. MR0967343 8

work page 1988
[4]

Buser, Geometry and spectra of compact Riemann surfaces , Progress in Mathematics, 106, Birkh¨ auser, Boston, MA, 1992

P. Buser, Geometry and spectra of compact Riemann surfaces , Progress in Mathematics, 106, Birkh¨ auser, Boston, MA, 1992. Zbl 1239.32001

work page arXiv 1992
[5]

Buser, Isospectral Riemann surfaces, Ann

P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier, 36 (1986), 167–192. MR0850750

work page 1986
[6]

J. H. Conway, N. J. A. Sloane, Four-dimensional lattices with the same theta series , Internat. Math. Res. Notices 4 (1992), 93–96. MR1159450

work page 1992
[7]

E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? , Linear Algebra and its Applications, 373 (2003), 241–272. MR2022290

work page 2003
[8]

A. G. Earnest, G. Nipp, On the theta series of positive quaternary quadratic forms , C. R. Math. Rep. Acad. Sci. Canada, 13:1 (1991), 33–38. MR1097501

work page 1991

Show all 28 references

[9]

Fiedler, Algebraic connectivity of graphs , Czech

M. Fiedler, Algebraic connectivity of graphs , Czech. Math. J., 23 (98) (1973) 298–305. MR0318007

1973
[10]

Friedli, The bundle Laplacian on discrete tori , Ann

F. Friedli, The bundle Laplacian on discrete tori , Ann. Inst. Henri Poincar´ e, Comb. Phys. Interact.6:1 (2019), pp. 97–121. MR3911691

2019
[11]

Godsil, D.A

C.D. Godsil, D.A. Holton, B. McKay, The Spectrum of a Graph . In: A. Dold, B. Eckmann, C.H.C Little (eds.), Lect. Notes. Math., 622, Springer, Berlin, Heidelberg, 1977. MR0544356

1977
[12]

Imrich, S

W. Imrich, S. Klavzar, Product graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscien ce, New York, 2000

2000
[13]

Isangulov, Isospectral ﬂat Klein bottles (Russian), Mat

R.R. Isangulov, Isospectral ﬂat Klein bottles (Russian), Mat. Zamet. YAGU, 7:2 (2000), 39–48. Zbl 0983.58016

2000
[14]

Kac, Can one hear the shape of a drum? , Amer

M. Kac, Can one hear the shape of a drum? , Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR201237

1966
[15]

Kitaoka, Positive deﬁnite quadratic forms with the same representat ion numbers, Arch

Y. Kitaoka, Positive deﬁnite quadratic forms with the same representat ion numbers, Arch. Math. (Basel), 28:5 (1977), 495–497. MR441864

1977
[16]

Kneser, Lineare Relationen zwischen Darstellungsanzahlen quadra tischer Formen (German), Math

M. Kneser, Lineare Relationen zwischen Darstellungsanzahlen quadra tischer Formen (German), Math. Ann., 168 (1967), 31–39. MR205943 9

1967
[17]

Y. Lin, S. Wan, H. Zhang, Connection Laplacian on discrete tori with converging prop erty, arXiv preprint arXiv:2403.06105, 2024 - arxiv.org

2024 arXiv
[18]

Algorithms Appl., 8:2 (2016) 1650028 (10 pages) MR3505475

Xiaogang Liu, Pengli Lu, Laplacian spectral characterization of dumbbell graphs an d theta graphs Discrete Math. Algorithms Appl., 8:2 (2016) 1650028 (10 pages) MR3505475

2016
[19]

Louis, Asymptotics for the determinant of the combinatorial Lapla cian on hypercubic lattices European J

J. Louis, Asymptotics for the determinant of the combinatorial Lapla cian on hypercubic lattices European J. Comb., 63 (2017), 176–196. MR3645793

2017
[20]

Mednykh, I

A. Mednykh, I. Mednykh, Isospectral genus two graphs are isomorphic , Ars Math. Contemp. 10:2 (2015), 223–235. MR3529288

2015
[21]

Milnor, Eigenvalues of the Laplace operator on certain manifolds , Proc

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds , Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR162204

1964
[22]

Mohar, The Laplacian spectrum of graphs , in Graph theory, combinatorics, and applications 2, Ed

B. Mohar, The Laplacian spectrum of graphs , in Graph theory, combinatorics, and applications 2, Ed. Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk, Wiley, New York (1991), 871– 898. MR1170831

1991
[23]

Nilsson, J

E. Nilsson, J. Rowlett, F. Rydell, The isospectral problem for ﬂat tori from three perspective s, Bull. Amer. Math. Soc. (New Series), 60:1 (2023), 39–83. MR4520776

2023
[24]

Sabidussi, Graph multiplication, Math

G. Sabidussi, Graph multiplication, Math. Z., 72 (1960), 446–457. (1963). MR0209177

1960
[25]

Schiemann, Ein Beispiel positiv deﬁniter quadratischer Formen der Dim ension 4 mit gleichen (German), Arch

A. Schiemann, Ein Beispiel positiv deﬁniter quadratischer Formen der Dim ension 4 mit gleichen (German), Arch. Math. (Basel), 54:4 (1990), 372–375. MR1042130

1990
[26]

Shiota, On theta series and the splitting of S2(Γ 0(q)), J

K.-i. Shiota, On theta series and the splitting of S2(Γ 0(q)), J. Math. Kyoto Univ., 31:4 (1991), 909–930. MR1141077

1991
[27]

V. G. Vizing, The Cartesian product of graphs (Russian). Vychisl. Sistemy, 9, (1963), 30–43. MR0209178 English translation in Comp. El. Syst. 2 (1966), 352–365

1963
[28]

Wolpert, The length spectra as moduli for compact Riemann surfaces , Ann

S. Wolpert, The length spectra as moduli for compact Riemann surfaces , Ann. Math. (2), 109:2 (1979), 323–351. MR0528966 10

1979

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Barden, H

D. Barden, H. Kang, Isospectral surfaces of genus two and three , Math. Proc. Camb. Phil. Soc., 153:1 (2012), 99 –110. MR2943668

work page 2012

[2] [2]

Brooks, R

R. Brooks, R. Tse, Isospectral surfaces of small genus , Nagoya Math. J. 107 (1987), 13–24; MR0909246 Corrigendum: R. Brooks, R. Tse, Na goya Math. J. 117 (1990), 227. MR1044942

work page 1987

[3] [3]

Brooks, Constructing isospectral manifolds , Amer

R. Brooks, Constructing isospectral manifolds , Amer. Math. Monthly, 95:9 (1988), 823–839. MR0967343 8

work page 1988

[4] [4]

Buser, Geometry and spectra of compact Riemann surfaces , Progress in Mathematics, 106, Birkh¨ auser, Boston, MA, 1992

P. Buser, Geometry and spectra of compact Riemann surfaces , Progress in Mathematics, 106, Birkh¨ auser, Boston, MA, 1992. Zbl 1239.32001

work page arXiv 1992

[5] [5]

Buser, Isospectral Riemann surfaces, Ann

P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier, 36 (1986), 167–192. MR0850750

work page 1986

[6] [6]

J. H. Conway, N. J. A. Sloane, Four-dimensional lattices with the same theta series , Internat. Math. Res. Notices 4 (1992), 93–96. MR1159450

work page 1992

[7] [7]

E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? , Linear Algebra and its Applications, 373 (2003), 241–272. MR2022290

work page 2003

[8] [8]

A. G. Earnest, G. Nipp, On the theta series of positive quaternary quadratic forms , C. R. Math. Rep. Acad. Sci. Canada, 13:1 (1991), 33–38. MR1097501

work page 1991