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