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Bakry–Émery Curvature Functions on Graphs
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 07 January 2019, pp. 89-143
- Print publication:
- February 2020
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- Article
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We study local properties of the Bakry–Émery curvature function
${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$ at a vertex 
$x$ of a graph 
$G$ systematically. Here 
${\mathcal{K}}_{G,x}({\mathcal{N}})$ is defined as the optimal curvature lower bound 
${\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality 
$CD({\mathcal{K}},{\mathcal{N}})$ that 
$x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and 
$S^{1}$-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 
$G_{1},G_{2}$ are equal to an abstract product of curvature functions of 
$G_{1},G_{2}$. 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 
$CD(0,\infty )$ but are not Cayley graphs.
