Non-negative discrete Bakry–Émery curvature implies polynomial growth

Let $G=(V,E)$ be a graph with bounded degrees. Write $\mathrm{dist}(x,y)$ for graph distance and define the open ball

$$B(x,r)=\{y\in V\colon \mathrm{dist}(x,y)<r\}.$$

Here $\mathrm{CD}(0,\infty)$ denotes the discrete Bakry–Émery curvature-dimension condition.

Non-negative curvature implies polynomial growth. If $G$ has bounded degrees and satisfies $\mathrm{CD}(0,\infty)$, then $G$ has polynomial growth. More precisely,

$$\forall x\in V,\quad \forall r\in\mathbb{N},\quad \#B(x,r)\le r^D,$$

where the constant $D<\infty$ depends only on the maximum degree.

This conjecture seeks the discrete analogue of the relationship between non-negative Bakry–Émery curvature and volume growth in Riemannian geometry. The general statement remains open, although it is established for edge-regular graphs and follows there from volume doubling.