Bakry-Émery curvature functions of graphs

Abstract

We study the Bakry-Émery 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-Émery 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.

In the new version, we carried out a slight restructuring of the material and added a concavity proof (in Prop. 4.1(iv)) thanks to the comments of Bobo Hua and Jim Portegies

References in corpus (1)

• Bakry-Emery curvature and diameter bounds on graphs

Cited by in corpus (4)

• Curvatures, graph products and Ricci flatness
• Ricci curvature, graphs and eigenvalues
• Condensed Ricci Curvature of Complete and Strongly Regular Graphs
• Perpetual cutoff method and discrete Ricci curvature bounds with exceptions