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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 |
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Vol. 13 (2020), No. 4, 559–576
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Abstract |
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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. |
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Keywords
coarse Ricci curvature, strongly regular graphs
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Mathematical Subject Classification 2010
Primary: 52C99, 53B99
Secondary: 05C10, 05C81, 05C99
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Milestones
Received: 26 August 2019
Revised: 11 March 2020
Accepted: 6 August 2020
Published: 20 November 2020
Communicated by Kenneth S. Berenhaut
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Authors |
| Vincent Bonini | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| Conor Carroll | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| Uyen Dinh | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| Sydney Dye | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| Joshua Frederick | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| Erin Pearse | |
| Mathematics Department California Polytechnic State University San Luis Obispo, CA United States |
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| © 2020 MSP (Mathematical Sciences Publishers). |
