Ricci curvature of graphs

Abstract

We 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. Introduction.The Ricci curvature plays a very important role on geometric analysis on Riemannian manifolds. Many results are established on manifolds with non-negative Ricci curvature or on manifolds with Ricci curvature bounded below.The definition of the Ricci curvature on metric spaces was first from the well-known Bakry and Emery notation. Bakry and Emery [1] found a way to define the "lower Ricci curvature bound" through the heat semigroup (P t ) t ≥0 on a metric measure space M. There are some recent works on giving a good notion for a metric measure space to have a "lower Ricci curvature bound", see [21], [18] and [19]. Those notations of Ricci curvature work on so called length spaces. In 2009, Ollivier [20] gave a notion of coarse Ricci curvature of Markov chains valid on arbitrary metric spaces, such as graphs.Graphs and manifolds are quite different in their nature. But they do share some similar properties through Laplace operators, heat kernels, and random walks, etc. Many pioneering works were done by Chung, Yau, and their coauthors [3,

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