Bakry-\'Emery curvature functions of graphs

Abstract

We study the Bakry-\'Emery curvature function $\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ of a vertex $x$ in a locally finite graph $G$ systematically. Here $\mathcal{K}_{G,x}(\mathcal{N})$ is defined as the optimal curvature lower bound $\mathcal{K}$ in the Bakry-\'Emery curvature-dimension inequality $CD(\mathcal{K},\mathcal{N})$ that $x$ satisfies. We prove the curvature functions of the Cartesian product of two graphs $G_1,G_2$ equal an abstract product of curvature functions of $G_1,G_2$. We relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We explore the curvature functions of Cayley graphs, strongly regular graphs, and many particular (families of) examples including Johnson graphs and complete bipartite graphs. We construct an infinite family of $6$-regular graphs which satisfy $CD(0,\infty)$ but are not Cayley graphs.