Bakry-\'Emery curvature functions of graphs

TL;DR

This paper systematically studies the Bakry-Émery curvature functions of graphs, exploring their properties, relationships with spectral graph theory, and analyzing specific graph families including Cayley and Johnson graphs.

Contribution

It introduces a systematic approach to analyze curvature functions of graphs, including their behavior under graph products and connections with spectral properties.

Findings

Curvature functions of Cartesian product graphs equal an abstract product of individual curvature functions.

Established relationships between curvature functions and spectral properties of graphs.

Constructed an infinite family of 6-regular graphs satisfying $CD(0,\infty)$ that are not Cayley 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.