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 on locally finite graphs. By using gradient estimate, We can get an estimate for the eigenvalue of Laplace operator on finite graphs. We also prove a Harnack inequality on finite graphs. As a consequence, we get a eigenvalue estimate for finite connected graphs with Ricci curvature bounded below by some constants which extends a similar result of Chung and Yau on Ricci flat graphs. We also modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.
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