Global Poincaré inequality on Graphs via Conical Curvature-Dimension Conditions

Abstract

We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs.

15 pages, minor corrections in v2 and generalized harmonic functions are getting their own manuscript so they have been removed from this version