{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T03:46:46Z","timestamp":1772250406466,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100012338","name":"Alan Turing Institute","doi-asserted-by":"publisher","award":["EPSRC grant EP\/N510129\/1"],"award-info":[{"award-number":["EPSRC grant EP\/N510129\/1"]}],"id":[{"id":"10.13039\/100012338","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is bounded both above and below by the generalized Cheeger constant, and the corresponding eigenfunctions can be used to approximate the Cheeger cut.<\/jats:p>","DOI":"10.1007\/s00373-021-02348-z","type":"journal-article","created":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T08:03:26Z","timestamp":1622880206000},"page":"2265-2286","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Cheeger Cut for Uniform Hypergraphs"],"prefix":"10.1007","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4995-6479","authenticated-orcid":false,"given":"Raffaella","family":"Mulas","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,5]]},"reference":[{"key":"2348_CR1","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0095-8956(85)90092-9","volume":"38","author":"N Alon","year":"1985","unstructured":"Alon, N., Milman, V.: $$\\lambda _1$$, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory B 38, 73\u201388 (1985)","journal-title":"J. Combin. Theory B"},{"key":"2348_CR2","unstructured":"Andreotti, E., Mulas, R.: Spectra of Signless Normalized Laplace Operators for Hypergraphs. arXiv:2005.14484"},{"key":"2348_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2020.01.012","author":"A Banerjee","year":"2020","unstructured":"Banerjee, A.: On the spectrum of hypergraphs. Linear Algebra Appl. (2020). https:\/\/doi.org\/10.1016\/j.laa.2020.01.012","journal-title":"Linear Algebra Appl."},{"issue":"4","key":"2348_CR4","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s11538-016-0158-0","volume":"78","author":"\u00c1 Bod\u00f3","year":"2016","unstructured":"Bod\u00f3, \u00c1., Katona, G., Simon, P.: SIS epidemic propagation on hypergraphs. Bull. Math. Biol. 78(4), 713\u2013735 (2016)","journal-title":"Bull. Math. Biol."},{"key":"2348_CR5","doi-asserted-by":"crossref","unstructured":"Buser, P.: On Cheeger\u2019s Inequality $$\\lambda _1\\ge h^2\/4$$. In: Geometry of the Laplace operator. Proceedings of Symposia in Pure Mathematics, Vol,. 6, pp. 29\u201377 (1980)","DOI":"10.1090\/pspum\/036\/573428"},{"issue":"2","key":"2348_CR6","first-page":"213","volume":"15","author":"P Buser","year":"1982","unstructured":"Buser, P.: A note on the isoperimetric constant. Ann. Sci. Ecol. Normal. Sup. Ser. IV 15(2), 213\u2013230 (1982)","journal-title":"Ann. Sci. Ecol. Normal. Sup. Ser. IV"},{"issue":"4","key":"2348_CR7","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1239\/aap\/1354716583","volume":"44","author":"AC Castro","year":"2012","unstructured":"Castro, A.C., Pelletier, B., Pudlo, P.: The normalized graph cut and cheeger constant: from discrete to continuous. Adv. Appl. Prob. 44(4), 907\u201393 (2012)","journal-title":"Adv. Appl. Prob."},{"key":"2348_CR8","doi-asserted-by":"publisher","DOI":"10.1145\/3178123","author":"T Chan","year":"2018","unstructured":"Chan, T., Louis, A., Tang, Z., Zhang, C.: Spectral properties of hypergraph Laplacian and approximation algorithms. J. ACM (2018). https:\/\/doi.org\/10.1145\/3178123","journal-title":"J. ACM"},{"issue":"5","key":"2348_CR9","doi-asserted-by":"publisher","first-page":"443","DOI":"10.4208\/jcm.1506-m2014-0164","volume":"33","author":"K Chang","year":"2015","unstructured":"Chang, K., Shao, S., Zhang, D.: The 1-laplacian cheeger cut: theory and algorithms. J. Comput. Math. 33(5), 443\u2013467 (2015)","journal-title":"J. Comput. Math."},{"key":"2348_CR10","doi-asserted-by":"crossref","unstructured":"Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, pp. 195\u2013199. Princeton University Press (1970)","DOI":"10.1515\/9781400869312-013"},{"key":"2348_CR11","volume-title":"Spectral graph theory","author":"F Chung","year":"1997","unstructured":"Chung, F.: Spectral graph theory. American Mathematical Society, New York (1997)"},{"issue":"2","key":"2348_CR12","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1090\/S0002-9947-1984-0743744-X","volume":"284","author":"J Dodziuk","year":"1984","unstructured":"Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787\u2013794 (1984)","journal-title":"Trans. Am. Math. Soc."},{"key":"2348_CR13","unstructured":"Ikeda, M., Miyauchi, A., Takai, Y., Yoshida, Y.: Finding Cheeger Cuts in Hypergraphs via Heat Equation. arXiv:1809.04396"},{"key":"2348_CR14","doi-asserted-by":"publisher","first-page":"870","DOI":"10.1016\/j.aim.2019.05.025","volume":"351","author":"J Jost","year":"2019","unstructured":"Jost, J., Mulas, R.: Hypergraph Laplace operators for chemical reaction networks. Adv. Math. 351, 870\u2013896 (2019)","journal-title":"Adv. Math."},{"issue":"5","key":"2348_CR15","doi-asserted-by":"publisher","first-page":"e1000385","DOI":"10.1371\/journal.pcbi.1000385","volume":"5","author":"S Klamt","year":"2009","unstructured":"Klamt, S., Haus, U., Theis, F.: Hypergraphs and cellular networks. PLoS Comp. Biol. 5(5), e1000385 (2009)","journal-title":"PLoS Comp. Biol."},{"issue":"1","key":"2348_CR16","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s10955-012-0543-5","volume":"151","author":"N Lanchier","year":"2013","unstructured":"Lanchier, N., Neufer, J.: Stochastic dynamics on hypergraphs and the spatial majority rule model. J. Stat. Phys. 151(1), 21\u201345 (2013)","journal-title":"J. Stat. Phys."},{"key":"2348_CR17","unstructured":"Li, P., Milenkovic, O.: Submodular hypergraphs: p-laplacians, cheeger inequalities and spectral clustering. In: Proceedings of the 35th International Conference on Machine Learning, PMLR, Vol. 80, pp. 3014\u20133023 (2018)"},{"issue":"4","key":"2348_CR18","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s11222-007-9033-z","volume":"17","author":"U von Luxburg","year":"2007","unstructured":"von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395\u2013416 (2007)","journal-title":"Stat. Comput."},{"issue":"2","key":"2348_CR19","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1214\/009053607000000640","volume":"36","author":"U von Luxburg","year":"2008","unstructured":"von Luxburg, U., Belkin, M., Bousquet, O.: Consistency of spectral clustering. Ann. Stat. 36(2), 555\u2013586 (2008). https:\/\/doi.org\/10.1214\/009053607000000640","journal-title":"Ann. Stat."},{"key":"2348_CR20","doi-asserted-by":"crossref","unstructured":"Mulas, R.: Sharp bounds for the largest eigenvalue of the normalized hypergraph Laplace Operator. Math. Notes (2021) (to appear)","DOI":"10.1134\/S0001434621010120"},{"issue":"1","key":"2348_CR21","doi-asserted-by":"publisher","first-page":"99","DOI":"10.2140\/astat.2020.11.99","volume":"11","author":"R Mulas","year":"2020","unstructured":"Mulas, R., Tran, N.: Minimal embedding dimensions of connected neural codes. Algebraic Stat. 11(1), 99\u2013106 (2020)","journal-title":"Algebraic Stat."},{"issue":"6","key":"2348_CR22","doi-asserted-by":"publisher","first-page":"112372","DOI":"10.1016\/j.disc.2021.112372","volume":"344","author":"R Mulas","year":"2021","unstructured":"Mulas, R., Zhang, D.: Spectral theory of Laplace Operators on oriented hypergraphs. Discrete Math. 344(6), 112372 (2021). https:\/\/doi.org\/10.1016\/j.disc.2021.112372","journal-title":"Discrete Math."},{"key":"2348_CR23","unstructured":"P\u00f3lya, G., Szeg\u0151, S.: Isoperimetric inequalities in mathematical physics. In: Annals of Mathematical Studies, Vol. 27. Princeton University Press, Princeton (1951)"},{"key":"2348_CR24","doi-asserted-by":"publisher","first-page":"2262","DOI":"10.1016\/j.laa.2012.06.011","volume":"437","author":"N Reff","year":"2012","unstructured":"Reff, N., Rusnak, L.: An oriented hypergraphic approach to algebraic graph theory. Linear Algebra Appl. 437, 2262\u20132270 (2012)","journal-title":"Linear Algebra Appl."},{"key":"2348_CR25","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/j.laa.2006.07.020","volume":"421","author":"D Spielman","year":"2007","unstructured":"Spielman, D., Teng, S.H.: Spectral partitioning works: planar graphs and finite element meshes. Linear Algebra Appl. 421, 284\u2013305 (2007)","journal-title":"Linear Algebra Appl."},{"key":"2348_CR26","unstructured":"Szlam, A., Bresson, X.: Total variation and Cheeger cuts. In: Proceedings of the 27th International Conference on International Conference on Machine Learning, ICML\u201910, pp. 1039\u20131046. Omnipress, Madison, WI (2010)"},{"issue":"10","key":"2348_CR27","doi-asserted-by":"publisher","first-page":"P10005","DOI":"10.1088\/1742-5468\/2010\/10\/P10005","volume":"2010","author":"Z Zhang","year":"2010","unstructured":"Zhang, Z., Liu, C.: A hypergraph model of social tagging networks. J. Stat. Mech. 2010(10), P10005 (2010)","journal-title":"J. Stat. Mech."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-021-02348-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-021-02348-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-021-02348-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,13]],"date-time":"2021-11-13T21:28:17Z","timestamp":1636838897000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-021-02348-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,5]]},"references-count":27,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["2348"],"URL":"https:\/\/doi.org\/10.1007\/s00373-021-02348-z","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"value":"0911-0119","type":"print"},{"value":"1435-5914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,5]]},"assertion":[{"value":"12 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that she has no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}