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Volumn 46, Issue 3, 2011, Pages 500-525

Curvature, Geometry and Spectral Properties of Planar Graphs

Author keywords

Cheeger's constant; Cut locus; Discrete curvature; Exponential growth; Hyperbolic properties; Spectrum of graphs; Unique continuation of eigenfunctions

Indexed keywords


EID: 80051825600     PISSN: 01795376     EISSN: 14320444     Source Type: Journal    
DOI: 10.1007/s00454-011-9333-0     Document Type: Article
Times cited : (51)

References (30)
  • 2
    • 0035585022 scopus 로고    scopus 로고
    • Curvature and geometry of tessellating plane graphs
    • Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25, 141-159 (2001).
    • (2001) Discrete Comput. Geom. , vol.25 , pp. 141-159
    • Baues, O.1    Peyerimhoff, N.2
  • 3
    • 33646765243 scopus 로고    scopus 로고
    • Geodesics in non-positively curved plane tessellations
    • Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243-263 (2006).
    • (2006) Adv. Geom. , vol.6 , Issue.2 , pp. 243-263
    • Baues, O.1    Peyerimhoff, N.2
  • 4
    • 45349101319 scopus 로고    scopus 로고
    • Gauss-Bonnet formula, Finiteness condition, and characterizations for graphs embedded in surfaces
    • Chen, B., Chen, G.: Gauss-Bonnet formula, Finiteness condition, and characterizations for graphs embedded in surfaces. Graphs Comb. 24(3), 159-183 (2008).
    • (2008) Graphs Comb. , vol.24 , Issue.3 , pp. 159-183
    • Chen, B.1    Chen, G.2
  • 5
    • 33644503891 scopus 로고    scopus 로고
    • A finitely presented group with unbounded dead-end depth
    • Erratum: Proc. Am. Math. Soc. 136(7), 2641-2645 (2008), see also arXiv:math/0406443
    • Cleary, S., Riley, T. R.: A finitely presented group with unbounded dead-end depth. Proc. Am. Math. Soc. 134(2), 343-349 (2006). Erratum: Proc. Am. Math. Soc. 136(7), 2641-2645 (2008), see also arXiv: math/0406443.
    • (2006) Proc. Am. Math. Soc. , vol.134 , Issue.2 , pp. 343-349
    • Cleary, S.1    Riley, T.R.2
  • 6
    • 0001398283 scopus 로고
    • Difference equations, isoperimetric inequalities and transience of certain random walks
    • Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc. 284, 787-794 (1984).
    • (1984) Trans. Am. Math. Soc. , vol.284 , pp. 787-794
    • Dodziuk, J.1
  • 7
    • 0001933963 scopus 로고
    • Spectral and function theory for combinatorial Laplacians
    • AMS Contemporary Mathematics
    • Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion. AMS Contemporary Mathematics, vol. 73, pp. 25-40 (1988).
    • (1988) Geometry of Random Motion , vol.73 , pp. 25-40
    • Dodziuk, J.1    Karp, L.2
  • 8
    • 0002908976 scopus 로고
    • Combinatorial Laplacians and isoperimetric inequality
    • Pitman Res. Notes Math. Ser
    • 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. 68-74 (1986).
    • (1986) From Local Times to Global Geometry, Control and Physics , vol.150 , pp. 68-74
    • Dodziuk, J.1    Kendall, W.S.2
  • 9
    • 38949196959 scopus 로고    scopus 로고
    • An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture
    • DeVos, M., Mohar, B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture. Trans. Am. Math. Soc. 359, 3287-3300 (2007).
    • (2007) Trans. Am. Math. Soc. , vol.359 , pp. 3287-3300
    • Devos, M.1    Mohar, B.2
  • 10
    • 28644443551 scopus 로고    scopus 로고
    • Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs
    • Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230, 184-221 (2006).
    • (2006) J. Funct. Anal. , vol.230 , pp. 184-221
    • Froese, R.1    Hasler, D.2    Spitzer, W.3
  • 11
    • 0037972746 scopus 로고    scopus 로고
    • Bochner's method for cell complexes and combinatorial Ricci curvature
    • Forman, R.: Bochner's method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323-374 (2003).
    • (2003) Discrete Comput. Geom. , vol.29 , Issue.3 , pp. 323-374
    • Forman, R.1
  • 12
    • 0038876918 scopus 로고    scopus 로고
    • Laplacians on rapidly branching trees
    • Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191-202 (1996).
    • (1996) Duke Math. J. , vol.83 , Issue.1 , pp. 191-202
    • Fujiwara, K.1
  • 14
    • 0000292564 scopus 로고
    • Random walks on graphs with a strong isoperimetric property
    • Gerl, P.: Random walks on graphs with a strong isoperimetric property. J. Theor. Probab. 1(2), 171-187 (1988).
    • (1988) J. Theor. Probab. , vol.1 , Issue.2 , pp. 171-187
    • Gerl, P.1
  • 15
    • 31244437181 scopus 로고    scopus 로고
    • Combinatorial curvature for planar graphs
    • Higuchi, Y.: Combinatorial curvature for planar graphs. Journal of Graph Theory (2001).
    • (2001) Journal of Graph Theory
    • Higuchi, Y.1
  • 16
    • 0035993055 scopus 로고    scopus 로고
    • Explicit isoperimetric constants and phase transitions in the random-cluster model
    • Häggström, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), 443-473 (2002).
    • (2002) Ann. Probab. , vol.30 , Issue.1 , pp. 443-473
    • Häggström, O.1    Jonasson, J.2    Lyons, R.3
  • 17
    • 62849103055 scopus 로고    scopus 로고
    • Isoperimetric constants of (d,f)-regular planar graphs
    • Higuchi, Y., Shirai, T.: Isoperimetric constants of (d, f)-regular planar graphs. Interdiscip. Inf. Sci. 9(2), 221-228 (2003).
    • (2003) Interdiscip. Inf. Sci. , vol.9 , Issue.2 , pp. 221-228
    • Higuchi, Y.1    Shirai, T.2
  • 18
    • 0242382528 scopus 로고    scopus 로고
    • Boundaries and random walks on finitely generated infinite groups
    • Karlsson, A.: Boundaries and random walks on finitely generated infinite groups. Ark. Mat. 41, 295-306 (2003).
    • (2003) Ark. Mat. , vol.41 , pp. 295-306
    • Karlsson, A.1
  • 19
    • 70350743416 scopus 로고    scopus 로고
    • Essential spectrum of the Laplacian on rapidly branching tessellations
    • Keller, M.: Essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51-66 (2010).
    • (2010) Math. Ann. , vol.346 , Issue.1 , pp. 51-66
    • Keller, M.1
  • 20
    • 79960115718 scopus 로고    scopus 로고
    • Cheeger constants, growth and spectrum of locally tessellating planar graphs
    • doi:10.1007/s00209-010-0699-0
    • Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. (2010). doi: 10. 1007/s00209-010-0699-0.
    • (2010) Math. Z.
    • Keller, M.1    Peyerimhoff, N.2
  • 21
    • 33646741992 scopus 로고    scopus 로고
    • Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature
    • 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), 1549-1559 (2005).
    • (2005) Proc. Am. Math. Soc. , vol.134 , Issue.5 , pp. 1549-1559
    • Klassert, S.1    Lenz, D.2    Peyerimhoff, N.3    Stollmann, P.4
  • 22
    • 38249030408 scopus 로고
    • Isoperimetric inequalities, growth, and the spectrum of graphs
    • Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119-131 (1988).
    • (1988) Linear Algebra Appl. , vol.103 , pp. 119-131
    • Mohar, B.1
  • 23
    • 0004418440 scopus 로고
    • Strongly geodesically automatic groups are hyperbolic
    • Papasoglu, P.: Strongly geodesically automatic groups are hyperbolic. Invent. Math. 121, 323-334 (1995).
    • (1995) Invent. Math. , vol.121 , pp. 323-334
    • Papasoglu, P.1
  • 24
    • 0001726384 scopus 로고
    • Recurrence and transience of the edge graph of a tiling of the Euclidean plane
    • Soardi, P. M.: Recurrence and transience of the edge graph of a tiling of the Euclidean plane. Math. Ann. 287, 613-626 (1990).
    • (1990) Math. Ann. , vol.287 , pp. 613-626
    • Soardi, P.M.1
  • 25
    • 84972539061 scopus 로고
    • A combinatorial analogue of a theorem of Myers
    • and Erratum: Illinois J. Math. 20(3), 551-554 (1976)
    • Stone, D. A.: A combinatorial analogue of a theorem of Myers. Ill. J. Math. 20(1), 12-21 (1976) and Erratum: Illinois J. Math. 20(3), 551-554 (1976).
    • (1976) Ill. J. Math. , vol.20 , Issue.1 , pp. 12-21
    • Stone, D.A.1
  • 26
    • 9244237547 scopus 로고    scopus 로고
    • Positively curved cubic plane graphs are finite
    • Sun, L., Yu, X.: Positively curved cubic plane graphs are finite. J. Graph Theory 47, 241-274 (2004).
    • (2004) J. Graph Theory , vol.47 , pp. 241-274
    • Sun, L.1    Yu, X.2
  • 27
    • 22444453472 scopus 로고    scopus 로고
    • A note on tilings and strong isoperimetric inequality
    • Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124, 385-393 (1998).
    • (1998) Math. Proc. Camb. Philos. Soc. , vol.124 , pp. 385-393
    • Woess, W.1
  • 28
    • 0002768694 scopus 로고
    • Random walks on infinite graphs and groups: a survey on selected topics
    • Woess, W.: Random walks on infinite graphs and groups: a survey on selected topics. Bull. Lond. Math. Soc. 26, 1-60 (1994).
    • (1994) Bull. Lond. Math. Soc. , vol.26 , pp. 1-60
    • Woess, W.1
  • 29
    • 69549090599 scopus 로고    scopus 로고
    • Heat kernel and essential spectrum of infinite graphs
    • Wojciechowski, R. K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419-1441 (2009).
    • (2009) Indiana Univ. Math. J. , vol.58 , pp. 1419-1441
    • Wojciechowski, R.K.1
  • 30
    • 0031316034 scopus 로고    scopus 로고
    • On the norms of the random walks on planar graphs
    • Żuk, A.: On the norms of the random walks on planar graphs. Ann. Inst. Fourier (Grenoble) 47(5), 1463-1490 (1997).
    • (1997) Ann. Inst. Fourier (Grenoble) , vol.47 , Issue.5 , pp. 1463-1490
    • Zuk, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.