On matrices associated to directed graphs and applications

This paper deals with the notions of 0-incidence and 1-incidence between edges on a directed graph associated to the line graph of a graph. The Laplacian energy and the signless Laplacian energy are obtained in a new way. From these results a relation between both energies is derived. Moreover, we obtain lower bounds for both the largest Laplacian eigenvalue and the largest signless Laplacian eigenvalue and prove that the latter is strictly greater than the first one.