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Discrete Curvature and Abelian Groups

  Published:2016-01-26
 Printed: Jun 2016
  • Bo'az Klartag,
    School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
  • Gady Kozma,
    Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
  • Peter Ralli,
    School of Mathematics and School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA
  • Prasad Tetali,
    School of Mathematics and School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA
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Abstract

We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of ``curvature'' in discrete spaces. An appealing feature of this discrete version of the so-called $\Gamma_2$-calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest -- particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs -- a result of independent interest.
Keywords: Ricci curvature, graph theory, abelian groups Ricci curvature, graph theory, abelian groups
MSC Classifications: 53C21, 57M15 show english descriptions Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Relations with graph theory [See also 05Cxx]
53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
57M15 - Relations with graph theory [See also 05Cxx]
 

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