Condensed Ricci curvature of complete and strongly regular graphs. (English) Zbl 1467.05277

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

Involve 13, No. 4, 559-576 (2020).

Summary: We study a modified notion of Ollivier’s coarse Ricci curvature on graphs introduced by Y. Lin et al. [Tohoku Math. J. (2) 63, No. 4, 605–627 (2011; Zbl 1237.05204)]. 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.

MSC:

05E30	Association schemes, strongly regular graphs
53B99	Local differential geometry
05C10	Planar graphs; geometric and topological aspects of graph theory
05C81	Random walks on graphs

Keywords:

coarse Ricci curvature; strongly regular graphs

Citations:

Zbl 1237.05204