A Note on the Order of Iterated Line Digraphs

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices nk of the k-iterated line digraph L (G), for k ≥ 0, where L(G) = G. We obtain this result by using the minimal polynomial of a quotient digraph π(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. Mathematics Subject Classifications: 05C20, 05C50.