Abstract
We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger’s constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of the essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies the absence of finitely supported eigenfunctions for nearest neighbor operators.
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Biggs, N.L., Mohar, B., Shawe-Taylor, J.: The spectral radius of infinite graphs. Bull. Lond. Math. Soc. 20(2), 116–120 (1988)
Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25, 141–159 (2001)
Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006)
Chen, B., Chen, G.: Gauss–Bonnet formula, Finiteness condition, and characterizations for graphs embedded in surfaces. Graphs Comb. 24(3), 159–183 (2008)
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
Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)
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)
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)
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)
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)
Forman, R.: Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323–374 (2003)
Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996)
Georgakopoulos, A.: A group has a flat Cayley complex if and only if it has a VAP-free Cayley graph. Preprint, arXiv:1011.4255
Gerl, P.: Random walks on graphs with a strong isoperimetric property. J. Theor. Probab. 1(2), 171–187 (1988)
Higuchi, Y.: Combinatorial curvature for planar graphs. Journal of Graph Theory (2001)
Häggström, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), 443–473 (2002)
Higuchi, Y., Shirai, T.: Isoperimetric constants of (d,f)-regular planar graphs. Interdiscip. Inf. Sci. 9(2), 221–228 (2003)
Karlsson, A.: Boundaries and random walks on finitely generated infinite groups. Ark. Mat. 41, 295–306 (2003)
Keller, M.: Essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010)
Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. (2010). doi:10.1007/s00209-010-0699-0
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)
Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119–131 (1988)
Papasoglu, P.: Strongly geodesically automatic groups are hyperbolic. Invent. Math. 121, 323–334 (1995)
Soardi, P.M.: Recurrence and transience of the edge graph of a tiling of the Euclidean plane. Math. Ann. 287, 613–626 (1990)
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)
Sun, L., Yu, X.: Positively curved cubic plane graphs are finite. J. Graph Theory 47, 241–274 (2004)
Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124, 385–393 (1998)
Woess, W.: Random walks on infinite graphs and groups: a survey on selected topics. Bull. Lond. Math. Soc. 26, 1–60 (1994)
Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419–1441 (2009)
Żuk, A.: On the norms of the random walks on planar graphs. Ann. Inst. Fourier (Grenoble) 47(5), 1463–1490 (1997)
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Keller, M. Curvature, Geometry and Spectral Properties of Planar Graphs. Discrete Comput Geom 46, 500–525 (2011). https://doi.org/10.1007/s00454-011-9333-0
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DOI: https://doi.org/10.1007/s00454-011-9333-0