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Можно услышать дискретный прямоугольный тор
Медных А. Д.1, Медных И. А.2, Соколова Г. К.3
1.1{Институт математики им. С.Л.Соболева, Новосибирск, Россия;
Дата поступления
2024.12.14		Аннотация. В работе показано, что два дискретных прямоугольных тора изоспектральны тогда и только тогда, когда они изоморфны.
Ключевые слова
матрица Лапласа, дискретный тор, изоспектральные многообразия, изоспектральные граф

Библиография
\bibitem{Milnor64} \textit{Milnor~J.} Eigenvalues of the Laplace operator on certain manifolds~// Proc. Nat. Acad. Sci. U.S.A. --~1964. --~Vol.~54. --~P.~542. --~MR162204v. --~DOI:~10.1073/pnas.51.4.542. \bibitem{Kac} \textit{Kac~M.} Can one hear the shape of a drum?~// Amer. Math. Monthly. --~1966. --~Vol.~73, No.~4.--~P.~1--23. --~MR201237. --~DOI:~10.2307/2313748. \bibitem{Wolpert} \textit{Wolpert~S.} The length spectra as moduli for compact Riemann surfaces~// Ann. Math. --~1979. --~Vol.~109, No.~2. --~P.~323--351. --~MR0528966. --~DOI:~10.2307/1971114. \bibitem{Buser86} \textit{Buser~P.} Isospectral Riemann surfaces~// Ann. Inst. Fourier. --~1986. --~Vol.~36. --~P.~167--192. --~MR0850750. --~URL: http://eudml.org/doc/74711. \bibitem{BrooksTse87} \textit{Brooks~R, Tse~R.} Isospectral surfaces of small genus~// Nagoya Math. --~J. 1990. --~Vol.~107. --~P.~13-24. --~MR0909246; Corrigendum: Brooks~R., Tse~R., Nagoya Math. J. --~1990. --~Vol.~117. --~P.~227. --~MR1044942. --~DOI:~10.1017/S0027763000002518. \bibitem{BardKang} \textit{Barden~D, Kang~H.} Isospectral surfaces of genus two and three~// Math. Proc. Camb. Phil. Soc. --~2012. --~Vol.~153, No.~1. --~P.~99-110. --~MR2943668. --~DOI:~10.1017/S0305004112000126. \bibitem{Brooks88} \textit{Brooks~R.} Constructing isospectral manifolds~// Amer. Math. Monthly. --~1988. --~Vol.~95, No.~9. --~P.~823--839. --~MR0967343. --~DOI:~10.1080/00029890.1988.11972094. \bibitem{Isang2000} \textit{Isangulov~R.R.} Isospectral flat Klein bottles (Russian)~// Mat. Zamet. YAGU. --~2000. --~Vol.~7, No.~2. --~P.~39--48. --~Zbl 0983.58016. \bibitem{Kneser} \textit{Kneser~M.} Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen~// Math. Ann. --~1967. --~Vol.~168. --~P.~31--39. --~MR205943. --~DOI:~10.1007/BF01361543. \bibitem{Kitaoka} \textit{Kitaoka~Y.} Positive definite quadratic forms with the same representation numbers~// Arch. Math. (Basel). --~1977. --~Vol.~28, No.~5. --~P.~495--497. --~MR441864. --~DOI:~10.1007/BF01223956. \bibitem{ConwaySloan} \textit{Conway~J.H., Sloane~N.J.A.} Four-dimensional lattices with the same theta series~// Internat. Math. Res. Notices. --~1992. --~Vol.~4. --~P.~93--96. --~MR1159450. --~DOI:~10.1155/S1073792892000102. \bibitem{Schiemann} \textit{Schiemann~A.} Ein Beispiel positiv definiter quadratischer Formen der Dimension $4$ mit gleichen Darstellungszahlen (German)~// Arch. Math. (Basel). --~1990. --~Vol.~54, --~No.~4. --~P.~372--375. --~MR1042130. --~DOI:~10.1007/BF01189584. \bibitem{Shiota} \textit{Shiota~K.-i.} On theta series and the splitting of $S_2(\Gamma_0(q))$~// J. Math. Kyoto Univ. --~1991. --~Vol.~31, No.~4. --~P.~909--930. --~MR1141077. --~DOI:~10.1215/kjm/1250519669. \bibitem{EarnNipp} \textit{Earnest~A. G., Nipp~G.} On the theta series of positive quaternary quadratic forms~// C. R. Math. Rep. Acad. Sci. Canada. --~1991. --~Vol.~13, No.~1. --~P.~33--38. --~MR1097501. %\href{https://mathreports.ca/article/on-the-theta-series-of-positive-quaternary-quadratic-forms/} \bibitem{DamHaemers} \textit{van Dam~E. R., Haemers~W. H.} Which graphs are determined by their spectrum?~// Linear Algebra and its Applications. --~2003. --~Vol.~373. --~P.~241--272. --~MR2022290. --~DOI:~10.1016/S0024-3795(03)00483-X. \bibitem{Buser92} \textit{Buser~P.} Geometry and spectra of compact Riemann surfaces. --~Birkh\"auser, Boston, MA~: Progress in Mathematics, 1992. --~Zbl 1239.32001. --~DOI:~10.1007/978-0-8176-4992-0. \bibitem{MedMedTheta} \textit{Mednykh~A., Mednykh~I.} Isospectral genus two graphs are isomorphic~// Ars Math. Contemp. --~2015. --~Vol.~10, No.~2. --~P.~223--235. --~MR3529288. --~DOI:~10.26493/1855-3974.550.e1a. \bibitem{LiuLu} \textit{Liu Xiaogang, Lu Pengli.} Laplacian spectral characterization of dumbbell graphs and theta graphs~// Discrete Math. Algorithms Appl. --~2016. --~Vol.~8, No.~2. --~1650028 (10 pages). --~MR3505475. --~DOI:~10.1142/S1793830916500282. \bibitem{NilRowRyd} \textit{Nilsson~E., Rowlett~J., Rydell~F.} The isospectral problem for flat tori from three perspectives~// Bull. Amer. Math. Soc. (New Series). --~2023. --~Vol.~60, No.~1. --~P.~39--83. --~MR4520776. --~DOI:~10.1090/bull/1770. \bibitem{Sabidu} \textit{Sabidussi~G. }Graph multiplication~// Math. Z. --~1960. --~Vol.~72. --~P.~446--457. --~MR0209177. --~DOI:~10.1007/BF01162967. \bibitem{Vizing} \textit{Vizing~V. G.} The Cartesian product of graphs (Russian)~// Vychisl. Sistemy. --~1963. --~Vol.~9. --~P.~30--43. --~MR0209178. English translation in Comp. El. Syst. --~1966. --~Vol.~2. --~P.~352--365. %\href{https://www.semanticscholar.org/paper/The-cartesian-product-of-graphs-Vizing/1af5fa6fd4cdb43baf9203d85015cae5eef2e5ea} \bibitem{ImriKlav} \textit{Imrich~W., Klavzar~S.} Product graph. --~Wiley-Interscience, New York~: Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000. %\href{https://www.amazon.com/Product-Graphs-Recognition-Wilfried-Imrich/dp/0471370398} \bibitem{Mohar91} \textit{Mohar~B.} The Laplacian spectrum of graphs // Graph theory, combinatorics, and applications / ed. Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk. --~1991. --~Vol.~2. --~P.~871--898. --~MR1170831. %\href{https://www.amazon.com/Graph-Theory-Combinatorics-Applications-Yousef/dp/0471532452} \bibitem{Fiedler} \textit{Fiedler~M.} Algebraic connectivity of graphs~// Czech. Math. J. --~1973. --~Vol.~23, No.~2.--~P.~298--305. --~MR0318007. %\href{https://eudml.org/doc/12723} \bibitem{Godsil} \textit{Godsil~C.D., Holton~D.A., McKay~B.} The Spectrum of a Graph // Dold~A., Eckmann~B., Little~C.H.C. (eds.) Lect. Notes. Math. --~Vol.~622. --~Springer, Berlin, Heidelberg, 1977. --~MR0544356. --~DOI:~10.1007/BFb0069184. \bibitem{Louis} \textit{Louis~J.} Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices~// European J. Comb. --~2017. --~Vol.~63. --~P.~176--196. --~MR3645793. --~DOI:~10.1016/j.ejc.2017.03.003. \bibitem{LinWanZhang} \textit{Lin~Y., Wan~S., Zhang~H.} Connection Laplacian on discrete tori with converging property. arXiv preprint arXiv:2403.06105, 2024 - arxiv.org, https://arxiv.org/abs/2403.06105. \bibitem{Friedli} \textit{Friedli~F.} The bundle Laplacian on discrete tori~// Ann. Inst. Henri Poincar\'e, Comb. Phys. Interact. --~2019. --~Vol.~6, No.~1. --~P.~97--121. --~MR3911691. --~DOI:~10.4171/AIHPD/66.

Сведения о финансировании и благодарности
The work is done in the framework by the State Contract of the Sobolev Institute of Mathematics (project no. FWNF--2022--0005).
One can hear a discrete rectangular torus
Mednykh A. D.1, Mednykh I. A.2, Sokolova G. K.3
1.1Sobolev Institute of Mathematics, Novosibirsk, Russia;
Received
2024.12.14		Abstract. In the present paper, we prove that two discrete rectangular tori are isospectral if and only if they are isomorphic.
Keywords
Laplacian matrix, discrete torus, isospectral manifolds, isospectral graphs

References
\bibitem{Milnor64} \textit{Milnor~J.} Eigenvalues of the Laplace operator on certain manifolds~// Proc. Nat. Acad. Sci. U.S.A. --~1964. --~Vol.~54. --~P.~542. --~MR162204v. --~DOI:~10.1073/pnas.51.4.542. \bibitem{Kac} \textit{Kac~M.} Can one hear the shape of a drum?~// Amer. Math. Monthly. --~1966. --~Vol.~73, No.~4.--~P.~1--23. --~MR201237. --~DOI:~10.2307/2313748. \bibitem{Wolpert} \textit{Wolpert~S.} The length spectra as moduli for compact Riemann surfaces~// Ann. Math. --~1979. --~Vol.~109, No.~2. --~P.~323--351. --~MR0528966. --~DOI:~10.2307/1971114. \bibitem{Buser86} \textit{Buser~P.} Isospectral Riemann surfaces~// Ann. Inst. Fourier. --~1986. --~Vol.~36. --~P.~167--192. --~MR0850750. --~URL: http://eudml.org/doc/74711. \bibitem{BrooksTse87} \textit{Brooks~R, Tse~R.} Isospectral surfaces of small genus~// Nagoya Math. --~J. 1990. --~Vol.~107. --~P.~13-24. --~MR0909246; Corrigendum: Brooks~R., Tse~R., Nagoya Math. J. --~1990. --~Vol.~117. --~P.~227. --~MR1044942. --~DOI:~10.1017/S0027763000002518. \bibitem{BardKang} \textit{Barden~D, Kang~H.} Isospectral surfaces of genus two and three~// Math. Proc. Camb. Phil. Soc. --~2012. --~Vol.~153, No.~1. --~P.~99-110. --~MR2943668. --~DOI:~10.1017/S0305004112000126. \bibitem{Brooks88} \textit{Brooks~R.} Constructing isospectral manifolds~// Amer. Math. Monthly. --~1988. --~Vol.~95, No.~9. --~P.~823--839. --~MR0967343. --~DOI:~10.1080/00029890.1988.11972094. \bibitem{Isang2000} \textit{Isangulov~R.R.} Isospectral flat Klein bottles (Russian)~// Mat. Zamet. YAGU. --~2000. --~Vol.~7, No.~2. --~P.~39--48. --~Zbl 0983.58016. \bibitem{Kneser} \textit{Kneser~M.} Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen~// Math. Ann. --~1967. --~Vol.~168. --~P.~31--39. --~MR205943. --~DOI:~10.1007/BF01361543. \bibitem{Kitaoka} \textit{Kitaoka~Y.} Positive definite quadratic forms with the same representation numbers~// Arch. Math. (Basel). --~1977. --~Vol.~28, No.~5. --~P.~495--497. --~MR441864. --~DOI:~10.1007/BF01223956. \bibitem{ConwaySloan} \textit{Conway~J.H., Sloane~N.J.A.} Four-dimensional lattices with the same theta series~// Internat. Math. Res. Notices. --~1992. --~Vol.~4. --~P.~93--96. --~MR1159450. --~DOI:~10.1155/S1073792892000102. \bibitem{Schiemann} \textit{Schiemann~A.} Ein Beispiel positiv definiter quadratischer Formen der Dimension $4$ mit gleichen Darstellungszahlen (German)~// Arch. Math. (Basel). --~1990. --~Vol.~54, --~No.~4. --~P.~372--375. --~MR1042130. --~DOI:~10.1007/BF01189584. \bibitem{Shiota} \textit{Shiota~K.-i.} On theta series and the splitting of $S_2(\Gamma_0(q))$~// J. Math. Kyoto Univ. --~1991. --~Vol.~31, No.~4. --~P.~909--930. --~MR1141077. --~DOI:~10.1215/kjm/1250519669. \bibitem{EarnNipp} \textit{Earnest~A. G., Nipp~G.} On the theta series of positive quaternary quadratic forms~// C. R. Math. Rep. Acad. Sci. Canada. --~1991. --~Vol.~13, No.~1. --~P.~33--38. --~MR1097501. %\href{https://mathreports.ca/article/on-the-theta-series-of-positive-quaternary-quadratic-forms/} \bibitem{DamHaemers} \textit{van Dam~E. R., Haemers~W. H.} Which graphs are determined by their spectrum?~// Linear Algebra and its Applications. --~2003. --~Vol.~373. --~P.~241--272. --~MR2022290. --~DOI:~10.1016/S0024-3795(03)00483-X. \bibitem{Buser92} \textit{Buser~P.} Geometry and spectra of compact Riemann surfaces. --~Birkh\"auser, Boston, MA~: Progress in Mathematics, 1992. --~Zbl 1239.32001. --~DOI:~10.1007/978-0-8176-4992-0. \bibitem{MedMedTheta} \textit{Mednykh~A., Mednykh~I.} Isospectral genus two graphs are isomorphic~// Ars Math. Contemp. --~2015. --~Vol.~10, No.~2. --~P.~223--235. --~MR3529288. --~DOI:~10.26493/1855-3974.550.e1a. \bibitem{LiuLu} \textit{Liu Xiaogang, Lu Pengli.} Laplacian spectral characterization of dumbbell graphs and theta graphs~// Discrete Math. Algorithms Appl. --~2016. --~Vol.~8, No.~2. --~1650028 (10 pages). --~MR3505475. --~DOI:~10.1142/S1793830916500282. \bibitem{NilRowRyd} \textit{Nilsson~E., Rowlett~J., Rydell~F.} The isospectral problem for flat tori from three perspectives~// Bull. Amer. Math. Soc. (New Series). --~2023. --~Vol.~60, No.~1. --~P.~39--83. --~MR4520776. --~DOI:~10.1090/bull/1770. \bibitem{Sabidu} \textit{Sabidussi~G. }Graph multiplication~// Math. Z. --~1960. --~Vol.~72. --~P.~446--457. --~MR0209177. --~DOI:~10.1007/BF01162967. \bibitem{Vizing} \textit{Vizing~V. G.} The Cartesian product of graphs (Russian)~// Vychisl. Sistemy. --~1963. --~Vol.~9. --~P.~30--43. --~MR0209178. English translation in Comp. El. Syst. --~1966. --~Vol.~2. --~P.~352--365. %\href{https://www.semanticscholar.org/paper/The-cartesian-product-of-graphs-Vizing/1af5fa6fd4cdb43baf9203d85015cae5eef2e5ea} \bibitem{ImriKlav} \textit{Imrich~W., Klavzar~S.} Product graph. --~Wiley-Interscience, New York~: Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000. %\href{https://www.amazon.com/Product-Graphs-Recognition-Wilfried-Imrich/dp/0471370398} \bibitem{Mohar91} \textit{Mohar~B.} The Laplacian spectrum of graphs // Graph theory, combinatorics, and applications / ed. Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk. --~1991. --~Vol.~2. --~P.~871--898. --~MR1170831. %\href{https://www.amazon.com/Graph-Theory-Combinatorics-Applications-Yousef/dp/0471532452} \bibitem{Fiedler} \textit{Fiedler~M.} Algebraic connectivity of graphs~// Czech. Math. J. --~1973. --~Vol.~23, No.~2.--~P.~298--305. --~MR0318007. %\href{https://eudml.org/doc/12723} \bibitem{Godsil} \textit{Godsil~C.D., Holton~D.A., McKay~B.} The Spectrum of a Graph // Dold~A., Eckmann~B., Little~C.H.C. (eds.) Lect. Notes. Math. --~Vol.~622. --~Springer, Berlin, Heidelberg, 1977. --~MR0544356. --~DOI:~10.1007/BFb0069184. \bibitem{Louis} \textit{Louis~J.} Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices~// European J. Comb. --~2017. --~Vol.~63. --~P.~176--196. --~MR3645793. --~DOI:~10.1016/j.ejc.2017.03.003. \bibitem{LinWanZhang} \textit{Lin~Y., Wan~S., Zhang~H.} Connection Laplacian on discrete tori with converging property. arXiv preprint arXiv:2403.06105, 2024 - arxiv.org, https://arxiv.org/abs/2403.06105. \bibitem{Friedli} \textit{Friedli~F.} The bundle Laplacian on discrete tori~// Ann. Inst. Henri Poincar\'e, Comb. Phys. Interact. --~2019. --~Vol.~6, No.~1. --~P.~97--121. --~MR3911691. --~DOI:~10.4171/AIHPD/66.

Acknowledgements
The work is done in the framework by the State Contract of the Sobolev Institute of Mathematics (project no. FWNF--2022--0005).
Сведения об авторах
Медных А. Д.
1.1. профессор{Институт математики им. С.Л.Соболева, Новосибирск, Россия
Адрес для корреспонденции:

Медных И. А.

Соколова Г. К.
About the authors
Mednykh A. D.
1.1. ProfessorSobolev Institute of Mathematics, Novosibirsk, Russia
Postal address:

Mednykh I. A.

Sokolova G. K.
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