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

TL;DR

This paper introduces the conical curvature-dimension condition for graphs, establishing its equivalence to sharp global Poincaré inequalities and deriving precise eigenvalue bounds and curvature estimates for complete graphs.

Contribution

It defines the novel conical curvature-dimension condition for graphs and links it to sharp inequalities and eigenvalue bounds, advancing geometric analysis on discrete structures.

Findings

Conical curvature-dimension condition characterizes sharp Poincaré inequalities.

Provides lower bounds for first eigenvalues of graphs.

Estimates the curvature of complete graphs precisely.

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 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.