Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions

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 Poincar\'e 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.