Bakry–Émery Curvature Functions on Graphs

We study local properties of the Bakry–Émery curvature function KG,x:(0,∞ ]→ R at a vertex x of a graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry–Émery curvature-dimension inequality CD(K,N) that x satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and S¹ -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 G₁,G₂ are equal to an abstract product of curvature functions of G₁,G₂ . 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.