  Given a digraph $G$, we propose a new method to find the recurrence equation
for the number of vertices $n_k$ of the $k$-iterated line digraph $L^k(G)$, for
$k\geq0$, where $L^0(G)=G$. We obtain this result by using the minimal
polynomial of a quotient digraph $\pi(G)$ of $G$. We show some examples of this
method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic
digraphs. In the first case, our method gives the enumeration of the ternary
length-2 squarefree words of any length.
