Vol. 13, No. 4, 2020

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Condensed Ricci curvature of complete and strongly regular graphs

Vincent Bonini, Conor Carroll, Uyen Dinh, Sydney Dye, Joshua Frederick and Erin Pearse

Vol. 13 (2020), No. 4, 559–576
Abstract

We study a modified notion of Ollivier’s coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than 1. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girths 4 and 5 using elementary means. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth 3 that is purely in terms of graph parameters.

Keywords
coarse Ricci curvature, strongly regular graphs
Mathematical Subject Classification 2010
Primary: 52C99, 53B99
Secondary: 05C10, 05C81, 05C99
Milestones
Received: 26 August 2019
Revised: 11 March 2020
Accepted: 6 August 2020
Published: 20 November 2020

Communicated by Kenneth S. Berenhaut
Authors
Vincent Bonini
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States
Conor Carroll
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States
Uyen Dinh
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States
Sydney Dye
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States
Joshua Frederick
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States
Erin Pearse
Mathematics Department
California Polytechnic State University
San Luis Obispo, CA
United States