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Global Poincar\'e inequality on Graphs via Conical Curvature-Dimension Conditions
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Global Poincar\'e 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 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.
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Cited by 1 Pith paper
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Nonnegative Bakry--\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\'e Inequalities
Connected bounded-degree graphs satisfying CD(0,∞) for the unnormalised Laplacian are volume doubling and satisfy scale-invariant L² Poincaré inequalities with dilation two.
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