Geodesic
On characteristical polinomial and eigenvectors in terms of tree-like structure of the graph
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 14-36

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While considering the square matrix as an adjacency matrix of a weighted digraph we construct an extended digraph, whose laplacian contains the original matrix as a submatrix. This construction allows us to use the known results on laplacians to study arbitrary square matrices. An eigenvector calculation in parametrical form demonstrates a connection between its components and a tree-like structure of the digraph. 
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     author = {V. A. Buslov},
     title = {On characteristical polinomial and eigenvectors in terms of tree-like structure of the graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {14--36},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_450_a1/}
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V. A. Buslov. On characteristical polinomial and eigenvectors in terms of tree-like structure of the graph. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 14-36. http://geodesic.mathdoc.fr/item/ZNSL_2016_450_a1/