1) How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain (Volume)?


2) How many nodes $v$ does a ball with center $x$ of radius $r$ in the Johnson graph $J(n,k)$ contain such that $d(x,v)=r$ (Surface)?


The Jonhson Graph is defined here:


<https://en.wikipedia.org/wiki/Johnson_graph>


Any reference to this question would be very nice.



How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain?
The number of vertices $p\_r$ which satisfy $d(x,v)=r$ in a distance regular graph can be computed from the intersection array:


$p\_r=\frac{b\_1b\_2...b\_r}{c\_1c\_2...c\_r}$


For Johnson graphs, $p\_r=(^k\_r)(^{n-k}\_{　r})$.


