Counting the spanning trees of a directed line graph

Abstract

The line graph LG of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix-Tree Theorem to prove a formula for the number of spanning trees of LG, and he asked for a bijective proof. In this paper, we give a bijective proof of a generating function identity due to Levine which generalizes Knuth's formula. As a result of this proof we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2^{n-1}. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine.