Bakry-Émery curvature on graphs as an eigenvalue problem

Cushing, David; Kamtue, Supanat; Liu, Shiping; Peyerimhoff, Norbert

doi:10.1007/s00526-021-02179-z

Bakry-Émery curvature on graphs as an eigenvalue problem

Cushing, David; Kamtue, Supanat; Liu, Shiping; Peyerimhoff, Norbert

Authors

David Cushing

Supanat Kamtue supanat.kamtue@durham.ac.uk
PGR Student Doctor of Philosophy

Shiping Liu

Professor Norbert Peyerimhoff norbert.peyerimhoff@durham.ac.uk
Professor

Abstract

In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

Citation

Cushing, D., Kamtue, S., Liu, S., & Peyerimhoff, N. (2022). Bakry-Émery curvature on graphs as an eigenvalue problem. Calculus of Variations and Partial Differential Equations, 61, Article 62. https://doi.org/10.1007/s00526-021-02179-z

Journal Article Type	Article
Acceptance Date	Dec 27, 2022
Online Publication Date	Feb 7, 2022
Publication Date	2022
Deposit Date	May 16, 2022
Publicly Available Date	Feb 7, 2023
Journal	Calculus of Variations and Partial Differential Equations
Print ISSN	0944-2669
Electronic ISSN	1432-0835
Publisher	Springer
Peer Reviewed	Peer Reviewed
Volume	61
Article Number	62
DOI	https://doi.org/10.1007/s00526-021-02179-z

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Copyright Statement
The version of record of this article, first published in Calculus of Variations and Partial Differential Equations, is available online at Publisher’s website: http://dx.doi.org/10.1007/s00526-021-02179-z