Bakry–Émery Curvature Functions on Graphs. (February 2020)

• Record Type: Journal Article
• Authors: Cushing, David; Liu, Shiping; Peyerimhoff, Norbert
• Is Part Of: Canadian journal of mathematics. Volume 72:Number 1(2020)
• Page Start: 89
• Page End: 143
• Publication Date: 2020-02
• Subjects: 05C50, -- 52C99, -- 53A40; Bakry–Emery curvature, -- curvature-dimension inequality, -- Cayley graph, -- Cartesian product; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2018-015-4
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: 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.