Abstract view
BakryÉmery Curvature Functions on Graphs


David Cushing,
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom
Shiping Liu,
School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China
Norbert Peyerimhoff,
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom
Abstract
We study local properties of the BakryÉmery curvature function
$\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ 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 BakryÉmery curvaturedimension
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.