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Bakry-Émery Curvature Functions on Graphs

Cushing, David; Liu, Shiping; Peyerimhoff, Norbert

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David Cushing

Shiping Liu


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.

Journal Article Type Article
Acceptance Date Apr 9, 2018
Online Publication Date Jul 5, 2018
Publication Date Feb 28, 2020
Deposit Date May 1, 2018
Publicly Available Date May 1, 2018
Journal Canadian Journal of Mathematics
Print ISSN 0008-414X
Electronic ISSN 1496-4279
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 72
Issue 1
Pages 89-143
Public URL


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