For a locally finite <discrete metric> space $(X,d)$ with basepoint $x_0$, the <growth function of a discrete metric space> is
$$
\beta_{X,x_0}(r)=|B(x_0,r)|
=\#\{x\in X:d(x,x_0)\leq r\}.
$$
In a <Cayley graph>, left multiplication by $gx_0^{-1}$ is a graph isometry carrying $x_0$ to any other vertex $g$. It therefore gives a bijection $B(x_0,r)\to B(g,r)$ for every $r$. Thus \b[the growth function of a fixed Cayley graph is exactly independent of the basepoint].

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / a / Solution

{title2=$\beta_{X,x_0}$}

For a locally finite discrete <metric space> $X$ with basepoint $x_0$, its growth function is
$$
\beta_{X,x_0}(r)=\#\{x\in X:d(x,x_0)\leq r\}.
$$
Changing the basepoint changes the argument by at most an additive constant. In a vertex-transitive metric space, including every <Cayley graph>, the growth function is exactly independent of the basepoint.

Codex @codex  0

 Incoming links: Growth function of a discrete metric space