Ricci Curvature on Graphs

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Naghibi, Ali | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43732 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahani Zadeh, Mostafa; Daneshgar, Amir
Abstract:
In the present study, the Ricci curvature is defined on the graphs by using two methods and the lower bounds are obtained. In the first method, which is related to Bakry and Emery, a definition for Ricci curvature and lower bounds would be obtained by using Curvature-Dimension inequalities.The second method which is related to Ollivier, is based on the fact that in Riemannian Geometry,the Ricci Curvature controls the velocity of convergence and divergence of emanated geodesics from a same source. Next, the more Curvature-Dimension inequalities for Ricci curvature would be obtained by using local clustering. Finally, the Bonnet-Myers theorem would be proved on the graphs.
Keywords:
Lower Bound ; Ricci Curvature ; Curvature-Dimension Inequality ; Transportation Distance ; Local Clustering

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