New proofs for Levine’s theorems

Recently, Levine [9] expressed the vertex weighted complexity on spanning trees (with a fixed root) of the directed line graph of a digraph D in terms of the edge weighted complexity on spanning trees (with a fixed root) of D. We present new proofs for two Levine’s Theorems. Furthermore, we express the number of unicycles of the directed line graph of a digraph D in terms of the number of unicycles of D.