Ricci curvature and eigenvalue estimate on locally finite graphs

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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Published 1 January 2010