Advertisement

A Frucht Theorem for Quantum Graphs

  • Delio MugnoloEmail author
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

A celebrated theorem due to R.F rucht states that, roughly speaking, each group is abstractly isomorphic to the symmetry group of some graph. By “symmetry group” the group of all graph automorphisms is meant. We provide an analogue of this result for quantum graphs, i.e., for Schrödinger equations on a metric graph, after suitably defining the notion of symmetry.

Keywords

Symmetries for evolution equations quantum graphs algebraic graph theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Babai. Automorphism group and category of cospectral graphs. Acta Math. Acad. Sci. Hung., 31: 295-306, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    L. Babai. Automorphism groups, isomorphism, reconstruction. In R.L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics - Vol. 2, pages 1447-1540. North-Holland, Amsterdam, 1995.Google Scholar
  3. 3.
    L. Babai. Private communication, 2010.Google Scholar
  4. 4.
    M. Behzad, G. Chartrand, and L. Lesniak-Foster. Graphs & Digraphs. Prindle, Weber & Schmidt, Boston, 1979.zbMATHGoogle Scholar
  5. 5.
    J. von Below. A characteristic equation associated with an eigenvalue problem on C2-networks. Lin. Algebra Appl., 71: 309-325, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    J. von Below. Can one hear the shape of a network? In F. Ali Mehmeti, J. von Below, and S. Nicaise, editors, Partial Differential Equations on Multistructures (Proc. Luminy 1999), volume 219 of Lect. Notes Pure Appl. Math., pages 19-36, New York, 2001. Marcel Dekker.Google Scholar
  7. 7.
    S. Cardanobile, D. Mugnolo, and R. Nittka. Well-posedness and symmetries of strongly coupled network equations. J. Phys. A, 41:055102, 2008.MathSciNetCrossRefGoogle Scholar
  8. 8.
    D.M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs - Theory and Applications. Pure Appl. Math. Academic Press, New York, 1979.Google Scholar
  9. 9.
    J. De Groot. Groups represented by homeomorphism groups I. Math. Ann., 138:80-102, 1959.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    R. Frucht. Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Math, 6:239-250, 1938.MathSciNetzbMATHGoogle Scholar
  11. 11.
    R. Frucht. Graphs of degree three with a given abstract group. Canadian J. Math,1:365-378, 1949.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S.A. Fulling, P. Kuchment, and J.H. Wilson. Index theorems for quantum graphs.J. Phys. A, 40:14165-14180, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    B. Gutkin and U. Smilansky. Can one hear the shape of a graph? J. Phys. A, 34:6061-6068, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    F. Harary and E.M. Palmer. On the point-group and line-group of a graph. Acta Math. Acad. Sci. Hung., 19:263-269, 1968.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    H. Izbicki. Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften. Monats. Math., 63:298-301, 1959.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    P. Kuchment. Quantum graphs: an introduction and a brief survey. In P. Exner, J. Keating, P. Kuchment, T. Sunada, and A. Teplyaev, editors, Analysis on Graphs and its Applications, volume 77 of Proc. Symp. Pure Math., pages 291-314, Providence, RI, 2008. Amer. Math. Soc.Google Scholar
  17. 17.
    J.W. Neuberger. Sobolev Gradients and Differential Equations, volume 1670 of Lect.Notes Math. Springer-Verlag, Berlin, 1997.Google Scholar
  18. 18.
    E.M. Ouhabaz. Analysis of Heat Equations on Domains, volume 30 of Lond. Math.Soc. Monograph Series. Princeton Univ. Press, Princeton, 2005.Google Scholar
  19. 19.
    Y.V. Pokornyi and A.V. Borovskikh. Differential equations on networks (geometric graphs). J. Math. Sci., 119:691-718, 2004.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    O. Post. First-order approach and index theorems for discrete and metric graphs.Ann. Henri Poincaré, 10:823-866, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G. Sabidussi. Graphs with given group and given graph-theoretical properties. Canad. J. Math, 9:515-525, 1957.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    G. Sabidussi. Graphs with given infinite group. Monats. Math., 64:64-67, 1960.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institut für AnalysisUniversität UlmUlmGermany

Personalised recommendations