  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 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 (a 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.
