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Bakry-Émery curvature on graphs as an eigenvalue problem

Cushing, David and Kamtue, Supanat and Liu, Shiping and Peyerimhoff, Norbert (2022) 'Bakry-Émery curvature on graphs as an eigenvalue problem.', Calculus of variations and partial differential equations., 61 . p. 62.

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.

Item Type:Article
Full text:Publisher-imposed embargo until 07 February 2023.
(AM) Accepted Manuscript
File format - PDF
(533Kb)
Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s00526-021-02179-z
Publisher 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
Date accepted:27 December 2022
Date deposited:16 May 2022
Date of first online publication:07 February 2022
Date first made open access:07 February 2023

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