On Cayley Graphs with Constant Ricci Curvature

Year 2024, Volume: 16 Issue: 1, 255 - 260, 30.06.2024

Mehmet Dağlı , Yonca Ünver

https://doi.org/10.47000/tjmcs.1492247

Abstract

Understanding the geometry of graphs has become increasingly important. One approach utilizes the Ricci curvature introduced by Lin, Lu, and Yau, which offers a valuable isomorphism invariant for locally finite graphs. One of the key tools used in calculating curvatures is the matching condition. This paper exploits the matching condition to construct families of Cayley graphs exhibiting constant Ricci curvature.

Keywords

Ricci curvature, Matching Condition, Cayley graph

References

• Bauer, F., Jost, J., Liu, L., Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator, Math. Res. Lett., 19(6)(2012), 1185–1205.
• Bhattacharya, B.B., Mukherjee, S., Exact and asymptotic results on coarse Ricci curvature of graphs, Discrete Math., 338(1)(2015), 23–42.
• Cushing, D., Kamtue, S., Kangaslampi, R., Liu, S., Münch, F. et al., Bakry-E´mery and Ollivier Ricci curvature of Cayley graphs, arXiv:2310.15953, (2023).
• Dağlı, M., Olmez, O., Smith, J.D.H., Ricci curvature, circulants, and extended matching conditions, B. Korean Math. Soc., 56(1)(2019), 201–217.
• Eidi, M., Jost, J., Ollivier Ricci curvature of directed hypergraphs, Scientific Reports, 10(2020).
• Li, H., Cao, J., Zhu, J., Liu, Y., Zhu, Q. et al. Curvature graph neural network, Information Sciences, 592(2022), 50–66.
• Lin, Y., Lu, L., Yau, S. T., Ricci curvature of graphs, Tohoku Math. J., 63(4)(2011), 605–627.
• Mizukai, I., Akifumi Sako, Ricci curvature of Cayley graphs for dihedral groups, generalized quaternion groups, and cyclic groups, arXiv:2210.00860v4, (2024).
• Ni, C.-C., Lin, Y.-Y., Gao, J., X. Gu, D. X., Saucan, E., Ricci curvature of the internet topology, IEEE Conference on Computer Communications (INFOCOM), Hong Kong, China (2015), 2758–2766.
• Ollivier, Y., Ricci curvature of Markov chains on metric spaces, Journal of Functional Analysis, 256(3)(2009), 810–864.
• Ollivier, Y., A survey of Ricci curvature for metric spaces and Markov chains, Advanced Studies in Pure Mathematics, 57(2010).
• Sandhu, R.S., Georgiou, T.T., Reznik, E., Zhu, L., Kolesov, I. et al., Graph curvature for differentiating cancer networks, Scientific Reports, 5(2015).
• Sandhu, R.S., Georgiou, T.T., Tannenbaum, A.R., Ricci curvature: An economic indicator for market fragility and systemic risk, Science Advances, 2(5)(2016).
• Smith, J.D.H., Ricci curvature, circulants, and a matching condition, Discrete Mathematics, 329(2014), 88–98.
• Ünver, Y., Cayley graphs with constant Ricci curvature, Master’s Thesis, Amasya University, 2022.