A Global Poincaré inequality on Graphs via a Conical Curvature-Dimension Condition

Sajjad Lakzian 1  and Zachary Mcguirk 2
  • 1 Mathematics Department, Fordham University, Rose Hill Campus Mathematics Department, Fordham University, Rose Hill Campus, , New York, USA
  • 2 Mathematics Department, Graduate Center, CUNY, , New York, USA

Abstract

We introduce and study the conical curvature-dimension condition, CCD(K, N), for finite graphs.We show that CCD(K, N) provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] N. Alon, O. Schwartz, and A. Shapira, An elementary construction of constant-degree expanders, Combin. Probab. Comput. 17 (2008), no. 3, 319-327.

  • [2] N. Alon and V. D. Milman, A1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), no. 1, 73-88.

  • [3] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206 (French).

  • [4] F. Bauer, M. Keller, and R.Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 2, 259-271.

  • [5] F. Bauer, F. Chung, Y. Lin, and Y. Liu, Curvature aspects of graphs, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2033-2042.

  • [6] F. Chung, Four proofs for the Cheeger inequality and graph partition algorithms, Fourth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math., vol. 48, Amer. Math. Soc., Providence, RI, 2010, pp. 331-349.

  • [7] , Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997.

  • [8] F. Chung, Y. Lin, and S.-T. Yau, Harnack inequalities for graphs with non-negative Ricci curvature, J. Math. Anal. Appl. 415 (2014), no. 1, 25-32.

  • [9] D. Cushing, S. Liu, and N. Peyerimhoff, Bakry-Émery curvature functions of graphs, ArXiv e-Print (2016). arXiv:1606.01496 [math.CO].

  • [10] D. Jozef, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787-794.

  • [11] M. Erbar, K. Kuwada, and K. T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), no. 3, 993-1071.

  • [12] M. Erbar, M. Rumpf, B. Schmitzer, and S. Simon, Computation of optimal transport on discrete metric measure spaces, ArXiv e-Print (2017). arXiv:1707.06859 [math.NA].

  • [13] J. Jürgen and S. Liu, Ollivier’s Ricci curvature, local clustering and curvature-dimension inequalities on graphs, Discrete Comput. Geom. 51 (2014), no. 2, 300-322.

  • [14] M. Keller and D. Mugnolo, General Cheeger inequalities for p-Laplacians on graphs, Nonlinear Anal. 147 (2016), 80-95.

  • [15] C. Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. (9) 103 (2015), no. 5, 1228-1275 (English, with English and French summaries).

  • [16] B. Klartag, G. Kozma, P. Ralli, and P. Tetali, Discrete curvature and abelian groups, Canad. J. Math. 68 (2016), no. 3, 655-674.

  • [17] Y. Lin, L. Lu, and S. - T. Yau, Ricci curvature of graphs, Tohoku Math. J. (2) 63 (2011), no. 4, 605-627.

  • [18] Y. Lin and S. - T. Yau, Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett. 17 (2010), no. 2, 343-356.

  • [19] S. Liu, F. Münch, and N. Peyerimhoff, Curvature and higher order Buser inequalities for the graph connection Laplacian, ArXiv e-Print (2015). arXiv:1512.08134 [math.SP].

  • [20] S. Liu and N. Peyerimhoff, Eigenvalue ratios of nonnegatively curved graphs, ArXiv e-Print (2014). arXiv:1406.6617 [math.SP].

  • [21] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903-991.

  • [22] , Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007), no. 1, 311-333.

  • [23] F. Münch, Remarks on curvature dimension conditions on graphs, Calc. Var. Partial Differential Equations 56 (2017), no. 1, Art. 11, 8.

  • [24] K. T. Sturm, A curvature-dimension condition for metric measure spaces, C. R. Math. Acad. Sci. Paris 342 (2006), no. 3, 197-200 (English, with English and French summaries).

  • [25] , On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65-131.

  • [26] , On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 65-131.

  • [27] R. Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)-City University of New York.

OPEN ACCESS

Journal + Issues

Analysis and Geometry in Metric Spaces (AGMS) is a fully peer-reviewed, open access electronic journal that publishes cutting-edge original research on analytical and geometrical problems in metric spaces and their mathematical applications. It features articles making connections among relevant topics in this field.

Search