Mathematical Research Letters

Volume 17 (2010)

Number 2

Ricci curvature and eigenvalue estimate on locally finite graphs

Pages: 343 – 356

DOI: http://dx.doi.org/10.4310/MRL.2010.v17.n2.a13

Authors

Yong Lin (Renmin University of China)

Shing-Tung Yau (Harvard University)

Abstract

We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by $-1$ on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: $$\lambda\ge {1\over d D(\exp( d D+1)-1)},$$ where $d$ is the weighted degree of $G$, and $D$ is the diameter of $G$.

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