Locating the Eigenvalues for Graphs of Small Clique-Width

  • Martin Fürer
  • Carlos Hoppen
  • David P. Jacobs
  • Vilmar Trevisan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


It is shown that if G has clique-width k, and a corresponding tree decomposition is known, then a diagonal matrix congruent to \(A - cI\) for constants c, where A is the adjacency matrix of the graph G of order n, can be computed in time \(O(k^2 n)\). This allows to quickly tell the number of eigenvalues in a given interval.


Eigenvalues Clique-width Congruent matrices Efficient algorithms Parameterized algorithms 


  1. 1.
    Alazemi, A., Andelić, M., Simić, S.K.: Eigenvalue location for chain graphs. Linear Algebra Appl. 505, 194–210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bıyıkoğlu, T., Simić, S.K., Stanić, Z.: Some notes on spectra of cographs. Ars Combin. 100, 421–434 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Braga, R.O., Rodrigues, V.M., Trevisan, V.: Locating eigenvalues of unicyclic graphs. Appl. Anal. Discrete Math. 11(2), 273–298 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformations, Volume 1: Foundations, pp. 313–400. World Scientific (1997)Google Scholar
  5. 5.
    Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci. 46(2), 218–270 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width minimization is NP-hard (extended abstract). In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC 2006, pp. 354–362. ACM, New York (2006)Google Scholar
  8. 8.
    Jacobs, D.P., Trevisan, V.: Locating the eigenvalues of trees. Linear Algebra Appl. 434(1), 81–88 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalue location in threshold graphs. Linear Algebra Appl. 439(10), 2762–2773 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jacobs, D.P., Trevisan, V., Tura, F.C.: Eigenvalue location in cographs. Discrete Appl. Math. (2017)Google Scholar
  11. 11.
    Janssens, D., Rozenberg, G.: On the structure of node-label-controlled graph languages. Inf. Sci. 20(3), 191–216 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Janssens, D., Rozenberg, G.: Restrictions, extensions, and variations of NLC grammars. Inf. Sci. 20(3), 217–244 (1980)CrossRefzbMATHGoogle Scholar
  13. 13.
    Johansson, Ö.: Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Congr. Numer. 132, 39–60 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kaminski, M., Lozin, V.V., Milanic, M.: Recent developments on graphs of bounded clique-width. Discrete Appl. Math. 157(12), 2747–2761 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). With 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual (iv+171 pp.)CrossRefGoogle Scholar
  16. 16.
    Mohammadian, A., Trevisan, V.: Some spectral properties of cographs. Discrete Math. 339(4), 1261–1264 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Royle, G.F.: The rank of a cograph. Electron. J. Combin. 10, Note 11, 7 pp. (electronic) (2003)Google Scholar
  20. 20.
    Sander, T.: On certain eigenspaces of cographs. Electron. J. Comb. 15(1), 8 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Stanić, Z.: On nested split graphs whose second largest eigenvalue is less than 1. Linear Algebra Appl. 430(8–9), 2200–2211 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wanke, E.: k-NLC graphs and polynomial algorithms. Discrete Appl. Math. 54(2), 251–266 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Martin Fürer
    • 1
  • Carlos Hoppen
    • 2
  • David P. Jacobs
    • 3
  • Vilmar Trevisan
    • 2
  1. 1.Department of Computer Science and EngineeringPennsylvania State UniversityState CollegeUSA
  2. 2.Instituto de MatemáticaUniversidade Federal do Rio Grande do SulAlegreBrazil
  3. 3.School of ComputingClemson UniversityClemsonUSA

Personalised recommendations