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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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/

[1] D. Cvetkovič, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application, VEB Deutscher Verlag der Wissenschlaften, Berlin, 1980 | MR | Zbl

[2] U. Tatt, Teoriya grafov, Mir, M., 1988

[3] P. Chebotarev, R. Agaev, “Forest matrices around the Laplacian matrix”, Linear Algebra and its Applications, 356 (2002), 253–274 | DOI | MR | Zbl

[4] M. Fiedler, J. Sedlaček, “O $w$-basich orientirovanych grafu”, Časopis Pěst. Mat., 83 (1958), 214–225 | MR | Zbl

[5] S. Chaiken, “A combinatorial proof of the all minors matrix tree theorem”, SIAM J. Algebraic Discrete Methods, 3:3 (1982), 319–329 | DOI | MR | Zbl

[6] J. W. Moon, “Some determinant expansions and the matrix-tree theorem”, Discrete Mathematics, 124 (1994), 163–171 | DOI | MR | Zbl

[7] V. A. Buslov, “O koeffitsientakh kharakteristicheskogo mnogochlena laplasiana vzveshennogo orientirovannogo grafa i teoreme o vsekh minorakh”, Zap. nauchn. sem. POMI, 427, 2014, 5–21

[8] A. D. Venttsel, M. I. Freidlin, Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmuschenii, M., 1979

[9] V. A. Buslov, Ierarkhiya markovskikh podprotsessov v modeli diskretnykh orientatsii, Kandidatskaya dissertatsiya, Sankt-Peterburg, 1992

[10] V. A. Buslov, K. A. Makarov, “Ierarkhiya masshtabov vremeni pri maloi diffuzii”, TMF, 76:2 (1988), 219–230 | MR | Zbl

[11] V. A. Buslov, K. A. Makarov, “Vremena zhizni i nizshie sobstvennye znacheniya operatora maloi diffuzii”, Mat. zametki, 51:1 (1992), 20–31 | MR | Zbl

[12] P. Yu. Chebotarëv, R. P. Agaev, Matrichnaya teorema o lesakh i laplasovskie matritsy orgrafov, LAP LAMBERT Academic Publising GmbH Co.Kg, 2011

[13] V. A. Buslov, M. S. Bogdanov, V. A. Khudobakhshov, “O minimalnom ostovnom dereve dlya orgrafov s potentsialnymi vesami”, Vestnik SPbGU, 10:3 (2008), 42–46