Let $G = (V,E)$ a graph, equipped with graph distance (i.e. for $x,y \in V$, the distance $d(x,y)$ is the length of the minimum path connecting $x$ and $y$). For $x \in V$ and $r \in \mathbb N$, define $B(x,r)$ as the ball centered ad $x$ with radius $r$:
$$ B(x,r) = \{ y \in V \ :\ d(x,y) \le r \} .$$


I would like to know if there is a common name for the following property, for a fixed integer $D$:


**D-dimensional growth**: the number of vertices in $B(x,r)$ scales at most as $(1+r)^D$:
$$ \exists c>0 \text{ s.t.} \quad \left| B(x,r) \right| \le c (1+r)^D \quad \forall r \in \mathbb N .$$


As a consequence, any such graph must have degree less than $c 2^D$. The converse is not true: take the binary tree as the simplest counter-example.


(Of course, any graph with $| V | \le 2^D$ will verify this property. But I am interested in "large" graphs, not in trivial cases.)



Graphs with polynomial volume growth
Yes, there is a common name for such graphs -- they are called *graphs with polynomial growth*. See e.g.


W. Imrich, N. Seifter: *A survey on graphs with polynomial growth*,
Discr. Math. 95 (1991), 101-117,
<http://www.math.uni-hamburg.de/home/diestel/books/directions/imrich-seifter.pdf>


