Bonini, Vincent; Carroll, Conor; Dinh, Uyen; Dye, Sydney; Frederick, Joshua; Pearse, Erin Condensed Ricci curvature of complete and strongly regular graphs. (English) Zbl 1467.05277 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. Cited in 1 Document 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 PDFBibTeX XMLCite \textit{V. Bonini} et al., Involve 13, No. 4, 559--576 (2020; Zbl 1467.05277) Full Text: DOI arXiv References: [1] 10.4310/MRL.2012.v19.n6.a2 · Zbl 1297.05143 · doi:10.4310/MRL.2012.v19.n6.a2 [2] 10.1073/pnas.43.9.842 · Zbl 0086.16202 · doi:10.1073/pnas.43.9.842 [3] 10.1016/j.disc.2014.08.012 · Zbl 1301.05078 · doi:10.1016/j.disc.2014.08.012 [4] 10.1137/17M1134469 · Zbl 1390.05048 · doi:10.1137/17M1134469 [5] ; Cameron, Graph theory, coding theory and block designs. London Math. Soc. Lecture Note Series, 19 (1975) · Zbl 0314.94008 [6] ; Chung, Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92 (1997) · Zbl 0867.05046 [7] 10.4153/cjm-2018-015-4 · Zbl 1430.05019 · doi:10.4153/cjm-2018-015-4 [8] 10.1147/rd.45.0497 · Zbl 0096.38102 · doi:10.1147/rd.45.0497 [9] 10.1007/s00454-013-9558-1 · Zbl 1294.05061 · doi:10.1007/s00454-013-9558-1 [10] 10.2748/tmj/1325886283 · Zbl 1237.05204 · doi:10.2748/tmj/1325886283 [11] 10.4310/CAG.2014.v22.n4.a3 · Zbl 1308.05038 · doi:10.4310/CAG.2014.v22.n4.a3 [12] 10.4007/annals.2009.169.903 · Zbl 1178.53038 · doi:10.4007/annals.2009.169.903 [13] 10.1016/j.aim.2019.106759 · Zbl 1426.82055 · doi:10.1016/j.aim.2019.106759 [14] 10.1016/j.jfa.2008.11.001 · Zbl 1181.53015 · doi:10.1016/j.jfa.2008.11.001 [15] 10.2969/aspm/05710343 · doi:10.2969/aspm/05710343 [16] 10.1016/j.disc.2014.04.010 · Zbl 1295.05187 · doi:10.1016/j.disc.2014.04.010 [17] 10.1007/s11511-006-0002-8 · Zbl 1105.53035 · doi:10.1007/s11511-006-0002-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.