Discrete curvature and abelian groups

Bo'az Klartag, Gady Kozma, Peter Ralli, Prasad Tetali

Research output: Contribution to journalArticlepeer-review

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 Γ2-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.

Original languageEnglish
Pages (from-to)655-674
Number of pages20
JournalCanadian Journal of Mathematics
Volume68
Issue number3
DOIs
StatePublished - Jun 2016

Keywords

  • Abelian groups
  • Graph theory
  • Ricci curvature

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