CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
  location:  PublicationsjournalsCJM
Abstract view

Bakry-Émery 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
Format:   LaTeX   MathJax  

Abstract

We study local properties of the Bakry-Émery 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 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.
Keywords: Bakry-Émery curvature, curvature-dimension inequality, Cayley graph, Cartesian product Bakry-Émery curvature, curvature-dimension inequality, Cayley graph, Cartesian product
MSC Classifications: 05C50, 52C99, 53A40 show english descriptions Graphs and linear algebra (matrices, eigenvalues, etc.)
None of the above, but in this section
Other special differential geometries
05C50 - Graphs and linear algebra (matrices, eigenvalues, etc.)
52C99 - None of the above, but in this section
53A40 - Other special differential geometries
 

© Canadian Mathematical Society, 2018 : https://cms.math.ca/