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Bakry-Émery curvature functions on graphs.

Cushing, David and Liu, Shiping and Peyerimhoff, Norbert (2020) 'Bakry-Émery curvature functions on graphs.', Canadian journal of mathematics., 72 (1). pp. 89-143.

Abstract

We study local properties of the Bakry-Émery curvature function KG,x:(0,∞]→R at a vertex x of a graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality CD(K,N) that x satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and S1-out regularity, and relate the curvature functions of G with various spectral properties of (weighted) graphs constructed from local structures of G. We prove that the curvature functions of the Cartesian product of two graphs G1,G2 are equal to an abstract product of curvature functions of G1,G2. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy CD(0,∞) but are not Cayley graphs.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.4153/CJM-2018-015-4
Publisher statement:This article has been published in a revised form in Canadian journal of mathematics http://doi.org/10.4153/CJM-2018-015-4. This version is published under a Creative Commons CC-BY-NC-ND. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © Canadian Mathematical Society 2018.
Date accepted:09 April 2018
Date deposited:01 May 2018
Date of first online publication:05 July 2018
Date first made open access:01 May 2018

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