  We calculate the distribution with respect to the vacuum state of the
distance-$k$ graph of a $d$-regular tree. From this result we show that the
distance-$k$ graph of a $d$-regular graphs converges to the distribution of the
distance-$k$ graph of a regular tree.
  Finally, we prove that, properly normalized, the asymptotic distributions of
distance-$k$ graphs of the $d$-fold free product graph, as $d$ tends to
infinity, is given by the distribution of $P_k(s)$, where $s$ is a semicircle
random variable and $P_k$ is the $k$-th Chebychev polynomial.
