Bakry–Émery Curvature Functions on Graphs. (February 2020) - Cushing, David; Liu, Shiping; Peyerimhoff, Norbert
Abstract: We study local properties of the Bakry–Émery curvature function ${\mathcal{K}}_{G, x}:(0, ∞ ]→ \mathbb{R}$ 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 $CD(0, ∞ )$ but are not Cayley graphs.