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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
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
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.