Journal of Mathematical Sciences

, Volume 236, Issue 5, pp 477–489 | Cite as

On the Relationship Between the Multiplicities of the Matrix Spectrum and the Signs of the Components of its Eigenvectors in a Tree-Like Structure

  • V. A. BuslovEmail author

We obtain a tree-like parametric representation of the eigenspace corresponding to an eigenvalue ⋋ of a matrix G in the case where the matrix G − ⋋E has a nonzero principal basic minor. If the algebraic and geometric multiplicities of ⋋ coincide, then such a minor always exists. The coefficients of powers of the spectral parameter are sums of terms of the same sign. If there is no nonzero principal basic minor, then the tree-like form does not allow one to represent the coefficients as sums of terms of the same sign, the only exception being the case of an eigenvalue of geometric multiplicity 1.


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  1. 1.
    D. Cvetkovič, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, VEB Deutscher Verlag der Wissenschlaften, Berlin (1980).zbMATHGoogle Scholar
  2. 2.
    G. Kirchhoff, “Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird,” Ann. Phys. Chem., 72, 497–508 (1847).CrossRefGoogle Scholar
  3. 3.
    S. Chaiken, “A combinatorial proof of the all minors matrix tree theorem,” SIAM J. Algebraic Discrete Methods, 3, No. 3, 319–329 (1982).MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. W. Moon, “Some determinant expansions and the matrix-tree theorem,” Discrete Math., 124, 163–171 (1994).MathSciNetCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    V. A. Buslov, “On the characteristic polynomial and eigenvectors in terms of the tree-line structure of a digraph,” Zap. Nauchn. Semin. POMI, 450, 14–36 (2016).Google Scholar
  7. 7.
    A. D. Ventsel and M. I. Freidlin, Fluctuations in Dynamical Systems under Small Random Perturbations [in Russian], Moscow (1979).Google Scholar
  8. 8.
    V. A. Buslov and K. A. Makarov, “Hierarchy of time scales in the case of weak diffusion,” Teoret. Mat. Fiz, 76, No. 2, 219–230 (1988).MathSciNetzbMATHGoogle Scholar
  9. 9.
    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).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Kelmans, I. Pak, and A. Postnikov, “Tree and forest volumes of graphs,” Rutcor Research Report, Rutgers Center for Operations Research, Rutgers University (1999).Google Scholar
  11. 11.
    P. Yu. Chebotarev and R. P. Agaev, Matrix Forest Theorem and Laplacian Matrices of Digraphs, Lambert Academic Publishers, Saarbr¨ucken (2011).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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