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On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph

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Regarding a square matrix as the 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 known results on Laplacians to study arbitrary square matrices. The calculation of an eigenvector in a parametric form demonstrates a connection between its components and the tree-like structure of the digraph.

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References

  1. D. Cvetkovič, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, Deutscher Verlag der Wissenschaften, Berlin (1980).

    MATH  Google Scholar 

  2. W. T. Tutte, Graph Theory, Addison-Wesley, Menlo Park (1988).

    MATH  Google Scholar 

  3. P. Chebotarev and R. Agaev, “Forest matrices around the Laplacian matrix,” Linear Algebra Appl., 356, 253–274 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Fiedler and J. Sedlaček, “O w-basich orientirovanych grafu,” Časopis Pěst. Mat., 83, 214–225 (1958).

    Google Scholar 

  5. S. Chaiken, “A combinatorial proof of the all minors matrix tree theorem,” SIAM J. Algebr. Discrete Methods, 3, No. 3, 319–329 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. W. Moon, “Some determinant expansions and the matrix-tree theorem,” Discrete Math., 124, 163–171 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. A. Buslov, “On the characteristic polynomial coefficients of the Laplace matrix of a weighted digraph and the all minors theorem,” Zap. Nauchn. Semin. POMI, 427, 5–21 (2014).

    Google Scholar 

  8. A. D. Ventsel and M. I. Freidlin, Fluctuations in Dynamical Systems Subject to Small Random Perturbations [in Russian], Moscow (1979).

  9. V. A. Buslov, “Hierarchy of Markov subprocesses in a model of discrete orientations,” PhD thesis, St.Petersburg (1992).

  10. V. A. Buslov and K. A. Makarov, “Hierarchy of time scales in the case of weak diffusion,” Teor. Mat. Fiz., 76, No. 2, 219–230 (1988).

    Article  MATH  Google Scholar 

  11. V. A. Buslov and K. A. Makarov, “Lifetimes and lower eigenvalues of an operator of small diffusion,” Mat. Zametki, 51, No. 1, 20–31 (1992).

    MathSciNet  MATH  Google Scholar 

  12. P. Yu. Chebotarev and R. P. Agaev, Matrix Forest Theorem and Laplacian Matrices of Digraphs, Lambert Academic Publishers, Saarbrücken (2011).

  13. V. A. Buslov, M. S. Bogdanov, and V. A. Khudobakshov, “On the minimum spanning tree for directed graphs with potential weights,” Vestnik St.Petersburg Univ., 10, No. 3, 42–46 (2008).

    Google Scholar 

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Authors and Affiliations

  1. St.Petersburg State University, St. Petersburg, Russia

    V. A. Buslov

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  1. V. A. Buslov
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Correspondence to V. A. Buslov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 14–36.

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Buslov, V.A. On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph. J Math Sci 232, 6–20 (2018). https://doi.org/10.1007/s10958-018-3854-5

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  • DOI: https://doi.org/10.1007/s10958-018-3854-5

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