On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg(+)(D)-A(D) and Q(D)=Deg(+)(D)+A(D), where A(D) represents the adjacency matrix and Deg(+)(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations.