  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.
