Geodesic
On characteristic polynomial coefficients of the Laplace matrix of a~weighted digraph and all minors theorem
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VII, Tome 427 (2014), pp. 5-21

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The simple proof of the expression of characteristic polynomial coefficients of the Laplace matrix of a weighted digraph in the form of sum over all incoming forests is submitted. The proof is based on the Laplace matrix expression as a product of weighted incidence matrices and investigation of relations between its minors and forests, which is useful to calculate all Laplace matrix minors. 
@article{ZNSL_2014_427_a0,
     author = {V. A. Buslov},
     title = {On  characteristic polynomial coefficients of the {Laplace} matrix of a~weighted digraph and all minors theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--21},
     publisher = {mathdoc},
     volume = {427},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_427_a0/}
}
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V. A. Buslov. On  characteristic polynomial coefficients of the Laplace matrix of a~weighted digraph and all minors theorem. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VII, Tome 427 (2014), pp. 5-21. http://geodesic.mathdoc.fr/item/ZNSL_2014_427_a0/