{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:51:36Z","timestamp":1772369496331,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2011,3,1]],"date-time":"2011-03-01T00:00:00Z","timestamp":1298937600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2011,10]]},"DOI":"10.1007\/s00454-011-9333-0","type":"journal-article","created":{"date-parts":[[2011,3,1]],"date-time":"2011-03-01T20:49:54Z","timestamp":1299012594000},"page":"500-525","source":"Crossref","is-referenced-by-count":31,"title":["Curvature, Geometry and Spectral Properties of Planar Graphs"],"prefix":"10.1007","volume":"46","author":[{"given":"Matthias","family":"Keller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2011,3,1]]},"reference":[{"issue":"2","key":"9333_CR1","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1112\/blms\/20.2.116","volume":"20","author":"N.L. Biggs","year":"1988","unstructured":"Biggs, N.L., Mohar, B., Shawe-Taylor, J.: The spectral radius of infinite graphs. Bull. Lond. Math. Soc. 20(2), 116\u2013120 (1988)","journal-title":"Bull. Lond. Math. Soc."},{"key":"9333_CR2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s004540010076","volume":"25","author":"O. Baues","year":"2001","unstructured":"Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25, 141\u2013159 (2001)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"9333_CR3","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1515\/ADVGEOM.2006.014","volume":"6","author":"O. Baues","year":"2006","unstructured":"Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243\u2013263 (2006)","journal-title":"Adv. Geom."},{"issue":"3","key":"9333_CR4","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s00373-008-0782-z","volume":"24","author":"B. Chen","year":"2008","unstructured":"Chen, B., Chen, G.: Gauss\u2013Bonnet formula, Finiteness condition, and characterizations for graphs embedded in surfaces. Graphs Comb. 24(3), 159\u2013183 (2008)","journal-title":"Graphs Comb."},{"issue":"2","key":"9333_CR5","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1090\/S0002-9939-05-08043-3","volume":"134","author":"S. Cleary","year":"2006","unstructured":"Cleary, S., Riley, T.R.: A finitely presented group with unbounded dead-end depth. Proc. Am. Math. Soc. 134(2), 343\u2013349 (2006). Erratum: Proc. Am. Math. Soc. 136(7), 2641\u20132645 (2008), see also arXiv:math\/0406443","journal-title":"Proc. Am. Math. Soc."},{"key":"9333_CR6","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1090\/S0002-9947-1984-0743744-X","volume":"284","author":"J. Dodziuk","year":"1984","unstructured":"Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc. 284, 787\u2013794 (1984)","journal-title":"Trans. Am. Math. Soc."},{"key":"9333_CR7","series-title":"AMS Contemporary Mathematics","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1090\/conm\/073\/954626","volume-title":"Geometry of Random Motion","author":"J. Dodziuk","year":"1988","unstructured":"Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion. AMS Contemporary Mathematics, vol. 73, pp. 25\u201340 (1988)"},{"key":"9333_CR8","series-title":"Pitman Res. Notes Math. Ser.","first-page":"68","volume-title":"From Local Times to Global Geometry, Control and Physics","author":"J. Dodziuk","year":"1986","unstructured":"Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: From Local Times to Global Geometry, Control and Physics. Pitman Res. Notes Math. Ser., vol. 150, pp. 68\u201374 (1986)"},{"key":"9333_CR9","doi-asserted-by":"crossref","first-page":"3287","DOI":"10.1090\/S0002-9947-07-04125-6","volume":"359","author":"M. DeVos","year":"2007","unstructured":"DeVos, M., Mohar, B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi\u2019s conjecture. Trans. Am. Math. Soc. 359, 3287\u20133300 (2007)","journal-title":"Trans. Am. Math. Soc."},{"key":"9333_CR10","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jfa.2005.04.004","volume":"230","author":"R. Froese","year":"2006","unstructured":"Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schr\u00f6dinger operators on graphs. J. Funct. Anal. 230, 184\u2013221 (2006)","journal-title":"J. Funct. Anal."},{"issue":"3","key":"9333_CR11","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s00454-002-0743-x","volume":"29","author":"R. Forman","year":"2003","unstructured":"Forman, R.: Bochner\u2019s method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323\u2013374 (2003)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"9333_CR12","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1215\/S0012-7094-96-08308-8","volume":"83","author":"K. Fujiwara","year":"1996","unstructured":"Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191\u2013202 (1996)","journal-title":"Duke Math. J."},{"key":"9333_CR13","unstructured":"Georgakopoulos, A.: A group has a flat Cayley complex if and only if it has a VAP-free Cayley graph. Preprint, arXiv:1011.4255"},{"issue":"2","key":"9333_CR14","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF01046933","volume":"1","author":"P. Gerl","year":"1988","unstructured":"Gerl, P.: Random walks on graphs with a strong isoperimetric property. J. Theor. Probab. 1(2), 171\u2013187 (1988)","journal-title":"J. Theor. Probab."},{"key":"9333_CR15","doi-asserted-by":"crossref","unstructured":"Higuchi, Y.: Combinatorial curvature for planar graphs. Journal of Graph Theory (2001)","DOI":"10.1002\/jgt.10004"},{"issue":"1","key":"9333_CR16","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1214\/aop\/1020107775","volume":"30","author":"O. H\u00e4ggstr\u00f6m","year":"2002","unstructured":"H\u00e4ggstr\u00f6m, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), 443\u2013473 (2002)","journal-title":"Ann. Probab."},{"issue":"2","key":"9333_CR17","first-page":"221","volume":"9","author":"Y. Higuchi","year":"2003","unstructured":"Higuchi, Y., Shirai, T.: Isoperimetric constants of (d,f)-regular planar graphs. Interdiscip. Inf. Sci. 9(2), 221\u2013228 (2003)","journal-title":"Interdiscip. Inf. Sci."},{"key":"9333_CR18","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF02390817","volume":"41","author":"A. Karlsson","year":"2003","unstructured":"Karlsson, A.: Boundaries and random walks on finitely generated infinite groups. Ark. Mat. 41, 295\u2013306 (2003)","journal-title":"Ark. Mat."},{"issue":"1","key":"9333_CR19","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s00208-009-0384-y","volume":"346","author":"M. Keller","year":"2010","unstructured":"Keller, M.: Essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51\u201366 (2010)","journal-title":"Math. Ann."},{"key":"9333_CR20","author":"M. Keller","year":"2010","unstructured":"Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. (2010). doi: 10.1007\/s00209-010-0699-0","journal-title":"Math. Z."},{"issue":"5","key":"9333_CR21","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1090\/S0002-9939-05-08103-7","volume":"134","author":"S. Klassert","year":"2005","unstructured":"Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, P.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. Am. Math. Soc. 134(5), 1549\u20131559 (2005)","journal-title":"Proc. Am. Math. Soc."},{"key":"9333_CR22","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/0024-3795(88)90224-8","volume":"103","author":"B. Mohar","year":"1988","unstructured":"Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119\u2013131 (1988)","journal-title":"Linear Algebra Appl."},{"key":"9333_CR23","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/BF01884301","volume":"121","author":"P. Papasoglu","year":"1995","unstructured":"Papasoglu, P.: Strongly geodesically automatic groups are hyperbolic. Invent. Math. 121, 323\u2013334 (1995)","journal-title":"Invent. Math."},{"key":"9333_CR24","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/BF01446917","volume":"287","author":"P.M. Soardi","year":"1990","unstructured":"Soardi, P.M.: Recurrence and transience of the edge graph of a tiling of the Euclidean plane. Math. Ann. 287, 613\u2013626 (1990)","journal-title":"Math. Ann."},{"issue":"1","key":"9333_CR25","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1215\/ijm\/1256050156","volume":"20","author":"D.A. Stone","year":"1976","unstructured":"Stone, D.A.: A combinatorial analogue of a theorem of Myers. Ill. J. Math. 20(1), 12\u201321 (1976) and Erratum: Illinois J. Math. 20(3), 551\u2013554 (1976)","journal-title":"Ill. J. Math."},{"key":"9333_CR26","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1002\/jgt.20026","volume":"47","author":"L. Sun","year":"2004","unstructured":"Sun, L., Yu, X.: Positively curved cubic plane graphs are finite. J. Graph Theory 47, 241\u2013274 (2004)","journal-title":"J. Graph Theory"},{"key":"9333_CR27","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1017\/S0305004197002429","volume":"124","author":"W. Woess","year":"1998","unstructured":"Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124, 385\u2013393 (1998)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"9333_CR28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/blms\/26.1.1","volume":"26","author":"W. Woess","year":"1994","unstructured":"Woess, W.: Random walks on infinite graphs and groups: a survey on selected topics. Bull. Lond. Math. Soc. 26, 1\u201360 (1994)","journal-title":"Bull. Lond. Math. Soc."},{"key":"9333_CR29","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1512\/iumj.2009.58.3575","volume":"58","author":"R.K. Wojciechowski","year":"2009","unstructured":"Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419\u20131441 (2009)","journal-title":"Indiana Univ. Math. J."},{"issue":"5","key":"9333_CR30","doi-asserted-by":"crossref","first-page":"1463","DOI":"10.5802\/aif.1606","volume":"47","author":"A. \u017buk","year":"1997","unstructured":"\u017buk, A.: On the norms of the random walks on planar graphs. Ann. Inst. Fourier (Grenoble) 47(5), 1463\u20131490 (1997)","journal-title":"Ann. Inst. Fourier (Grenoble)"}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-011-9333-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-011-9333-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-011-9333-0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T22:36:05Z","timestamp":1560033365000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-011-9333-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["9333"],"URL":"https:\/\/doi.org\/10.1007\/s00454-011-9333-0","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3,1]]}}}