Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Inverse Eigenvalue Problem for Constructing a Kind of Acyclic Matrices with Two Eigenpairs

  • 1 Accesses

Abstract

We investigate an inverse eigenvalue problem for constructing a special kind of acyclic matrices. The problem involves the reconstruction of the matrices whose graph is an m-centipede. This is done by using the (2m − 1)st and (2m)th eigenpairs of their leading principal submatrices. To solve this problem, the recurrence relations between leading principal submatrices are used.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    E. Andrade, H. Gomes, M. Robbiano: Spectra and Randić spectra of caterpillar graphs and applications to the energy. MATCH Commun. Math. Comput. Chem. 77 (2017), 61–75.

  2. [2]

    C. Bu, J. Zhou, H. Li: Spectral determination of some chemical graphs. Filomat 26 (2012), 1123–1131.

  3. [3]

    M. T. Chu, G. H. Golub: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2005.

  4. [4]

    A. L. Duarte: Construction of acyclic matrices from spectral data. Linear Algebra Appl. 113 (1989), 173–182.

  5. [5]

    S. Elhay, G. M. L. Gladwell, G. H. Golub, Y. M. Ram: On some eigenvector-eigenvalue relations. SIAM J. Matrix Anal. Appl. 20 (1999), 563–574.

  6. [6]

    K. Ghanbari, F. Parvizpour: Generalized inverse eigenvalue problem with mixed eigen-data. Linear Algebra Appl. 437 (2012), 2056–2063.

  7. [7]

    L. Hogben: Spectral graph theory and the inverse eigenvalue problem of a graph. Electron. J. Linear Algebra 14 (2005), 12–31.

  8. [8]

    K. H. Monfared, B. L. Shader: Construction of matrices with a given graph and prescribed interlaced spectral data. Linear Algebra Appl. 438 (2013), 4348–4358.

  9. [9]

    R. Nair, B. L. Shader: Acyclic matrices with a small number of distinct eigenvalues. Linear Algebra Appl. 438 (2013), 4075–4089.

  10. [10]

    P. Nylen, F. Uhlig: Inverse eigenvalue problems associated with spring-mass systems. Linear Algebra Appl. 254 (1997), 409–425.

  11. [11]

    J. Peng, X.-Y. Hu, L. Zhang: Two inverse eigenvalue problems for a special kind of matrices. Linear Algebra Appl. 416 (2006), 336–347.

  12. [12]

    H. Pickmann, J. Egaña, R. L. Soto: Extremal inverse eigenvalue problem for bordered diagonal matrices. Linear Algebra Appl. 427 (2007), 256–271.

  13. [13]

    V. Pivovarchik, N. Rozhenko, C. Tetter: Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings. Linear Algebra Appl. 439 (2013), 2263–2292.

  14. [14]

    M. Sen, D. Sharma: Generalized inverse eigenvalue problem for matrices whose graph is a path. Linear Algebra Appl. 446 (2014), 224–236.

  15. [15]

    D. Sharma, M. Sen: Inverse eigenvalue problems for two special acyclic matrices. Mathematics 4 (2016), Article ID 12, 11 pages.

  16. [16]

    D. Sharma, M. Sen: Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede. Spec. Matrices 6 (2018), 77–92.

  17. [17]

    Y. Zhang: On the general algebraic inverse eigenvalue problems. J. Comput. Math. 22 (2004), 567–580.

Download references

Author information

Correspondence to Seyed Abolfazl Shahzadeh Fazeli.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zarch, M.B., Fazeli, S.A.S. & Karbassi, S.M. Inverse Eigenvalue Problem for Constructing a Kind of Acyclic Matrices with Two Eigenpairs. Appl Math 65, 89–103 (2020). https://doi.org/10.21136/AM.2020.0103-19

Download citation

Keywords

  • inverse eigenvalue problem
  • leading principal submatrices
  • graph of a matrix
  • eigenpair

MSC 2010

  • 65F18
  • 05C50