Combinatorial explanation of coefficients of the signless Laplacian characteristic polynomial of a digraph

Let G be a simple digraph with n vertices $v_1,v_2,\dots ,v_n$. Denote the adjacency matrix and the in-degree matrix of G by $A={\left(a_{ij}\right)}_{n\times n}$ and $D=diag(d_1^{+},d_2^{+},\cdots ,d_n^{+})$, respectively, where $a_{ij}=1$ if $(v_i,v_j)$ is an arc of G and $a_{ij}=0$ otherwise, and $d_i^{+}$ is the number of arcs with head $v_i$ in G. Set $f(G;x)=\det (x{I}_n-D-A)=\sum _{i=0}^n{(-1)}^i{c}_i{x}^{n-i}$, where $\det \left(X\right)$ denotes the determinant of a square matrix X. Then $f(G;x)$ is called the signless Laplacian characteristic polynomial of the digraph G. Li, Lu, Wang and Wang (2023) [7] gave a combinatorial explanation of $c_{n-1}$ of $f(G;x)$. In this paper, we give combinatorial explanations of all the coefficients of $f(G;x)$ and we also generalize this result to weighted digraphs.