Advertisement

Ukrainian Mathematical Journal

, Volume 64, Issue 3, pp 325–343 | Cite as

Strongly connected simply laced quivers and their eigenvectors

  • I. V. Dudchenko
  • V. V. Kirichenko
  • M. V. Plakhotnyk
Article
  • 48 Downloads

We study the relationship between the isomorphism of quivers and properties of their spectra. It is proved that strongly connected simply laced quivers with at most four vertices are isomorphic to one another if and only if their characteristic polynomials coincide and their left and right normalized positive eigenvectors that correspond to the index can be obtained from one another by the permutation of their coordinates. An example showing that this statement is not true for quivers with five vertices is given.

Keywords

Adjacency Matrix Characteristic Polynomial Simple Root Dynkin Diagram Permutation Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).Google Scholar
  2. 2.
    I. V. Dudchenko, Strongly Connected Quivers, Their Indices, and Eigenvectors [in Russian], Preprint, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).Google Scholar
  3. 3.
    I. V. Dudchenko and M. V. Plakhotnyk, “Strongly connected quivers and their eigenvectors,” in: Abstracts of the Fourth Conference of Young Scientists on Contemporary Problems of Mechanics and Mathematics (May 24–27, 2011, Lviv) [in Ukrainian] (2011), pp. 265–266.Google Scholar
  4. 4.
    F. Harary, Graph Theory, Addison-Wesley, London (1969).Google Scholar
  5. 5.
    P. Gabriel, “Unserlegbare Darstellungen 1,” Manuscr. Math., 6, 71–103 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    I. Dudchenko and M. Plakhotnyk, “A linear algorithm of checking of the graph connectness,” Alg. Discr. Math., 13, No. 1, 43–51 (2012).Google Scholar
  7. 7.
    M. Hazewinkel, N. Gubareni, and V. V. Kirichenko, Algebras, Rings and Modules, Kluwer (2004).Google Scholar
  8. 8.
    G. Frobenius, “Ü ber Matrizen aus positiven Elementen,” Sitzungsber. Akad. Wiss. Phys.-Math. Kl., 471–476 (1908), 514–518 (1909).Google Scholar
  9. 9.
    G. Frobenius, “Ü ber Matrizen aus nicht-negativen Elementen,” Sitzungsber. Akad. Wiss. Phys.-Math. Kl., 456–477 (1912).Google Scholar
  10. 10.
    M. A. Dokuchaev, N. M. Gubareni, V. M. Futorny, M. A. Khibina, and V. V. Kirichenko, Dynkin Diagrams and Spectra of Graphs, Preprint, Brazil (2010).Google Scholar
  11. 11.
    O. Perron, “Jacobischer Kettenbruchalgorithmus,” Math. Ann., 64, 1–76 (1907).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    O. Perron, “Ueber Matrizen,” Math. Ann., 64, 248–263 (1907).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • I. V. Dudchenko
    • 1
  • V. V. Kirichenko
    • 2
  • M. V. Plakhotnyk
    • 2
  1. 1.Slavyansk State Pedagogic UniversitySlavyanskUkraine
  2. 2.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations