A Global Poincaré inequality on Graphs via a Conical Curvature-Dimension Condition

Abstract

We introduce and study the conical curvature-dimension condition, CCD(K, N), for finite 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

Keywords

• graph
• curvature-dimension
• cone
• poincaré inequality
• ricci curvature
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