Discrete Curvature and Abelian Groups
Following the so-called gamma calculus of Bakry-Emery, we study a discrete Bochner-type inequality on graphs, and explore it as a notion of curvature in discrete spaces. We find that it is fairly straightforward to compute the curvature parameter. In particular, we perform the computation for several 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, relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger isoperimetric constant in terms of the spectral gap, for graphs with nonnegative curvature. A result of interest is that the tight bound on the Cheeger constant holds for Cayley graphs of finitely generated abelian groups.