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Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

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Abstract

For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization for strongly connected directed graphs by using the mean transition probability kernel which appears in the formulation of the Chung Laplacian. We conclude several geometric and spectral properties under a lower Ricci curvature bound extending previous results in the undirected case.

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Acknowledgements

The authors are grateful to the anonymous referees for valuable comments. The first author was supported in part by JSPS KAKENHI (19K14532). The first and second authors were supported in part by JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric Analysis for Materials Design” (17H06460).

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Authors and Affiliations

  1. Advanced Institute for Materials Research (AIMR), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Ryunosuke Ozawa & Yohei Sakurai

  2. Research Institute for Humanity and Nature, 457-4 Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8047, Japan

    Taiki Yamada

Authors
  1. Ryunosuke Ozawa
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  2. Yohei Sakurai
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  3. Taiki Yamada
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Correspondence to Yohei Sakurai.

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Communicated by J. Jost.

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Ozawa, R., Sakurai, Y. & Yamada, T. Geometric and spectral properties of directed graphs under a lower Ricci curvature bound. Calc. Var. 59, 142 (2020). https://doi.org/10.1007/s00526-020-01809-2

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