Bounded-degree polynomial growth conjecture for nonnegative Bakry–Émery graphs

Let $G=(V,E)$ be a graph with degree bound $d_*$, and let $B(x,r)$ denote the open ball of radius $r$ about $x$. Assume the degree and curvature hypotheses referred to as Assumptions~ and~. Bounded-degree polynomial growth conjecture. There is a finite exponent $D(d_*)$ such that

$$\#B(x,r)\leq r^{D(d_*)}$$

for every $x\in V$ and every integer $r\geq1$. The surrounding discussion presents this as the general bounded-degree case of the classical dimension-free conjecture; the cited edge-regular case is known, while the arbitrary bounded-degree case is described as remaining open.