Advertisement

Czechoslovak Mathematical Journal

, Volume 63, Issue 1, pp 73–90 | Cite as

Relations between (κ, τ)-regular sets and star complements

  • Milica AnđelićEmail author
  • Domingos M. Cardoso
  • Slobodan K. Simić
Article

Abstract

Let G be a finite graph with an eigenvalue µ of multiplicity m. A set X of m vertices in G is called a star set for µ in G if µ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (κ, τ)-regular if it induces a κ-regular subgraph and every vertex not in the subset has τ neighbors in it. We investigate the graphs having a (κ, τ)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.

Keywords

eigenvalue star complement non-main eigenvalue Hamiltonian graph 

MSC 2010

05C50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F.K. Bell: Characterizing line graphs by star complements. Linear Algebra Appl. 296 (1999), 15–25.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    F.K. Bell, S.K. Simić: On graphs whose star complement for −2 is a path or cycle. Linear Algebra Appl. 377 (2004), 249–265.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    D.M. Cardoso, D. Cvetković: Graphs with least eigenvalue −2 attaining a convex quadratic upper bound for the stability number. Bull., Cl. Sci. Math. Nat., Sci. Math. 133 (2006), 41–55.zbMATHCrossRefGoogle Scholar
  4. [4]
    D.M. Cardoso, P. Rama: Equitable bipartitions of graphs and related results. J. Math. Sci., New York 120 (2004), 869–880.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D.M. Cardoso, P. Rama: Spectral results on regular graphs with (k, τ)-regular sets. Discrete Math. 307 (2007), 1306–1316.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    D.M. Cardoso, P. Rama: Spectral results on graphs with regularity constraints. Linear Algebra Appl. 423 (2007), 90–98.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    D.M. Cardoso, I. Sciriha, C. Zerafa: Main eigenvalues and (k, τ)-regular sets. Linear Algebra Appl. 423 (2010), 2399–2408.MathSciNetCrossRefGoogle Scholar
  8. [8]
    D. Cvetković, M. Doob, H. Sachs: Spectra of Graphs. Theory and Application. Pure and Applied Mathematics 87, Academic Press, New York, and VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.Google Scholar
  9. [9]
    D. Cvetković, P. Rowlinson, S. Simić: Eigenspaces of Graphs. [Paperback reprint of the hardback edition 1997]. Encyclopedia of Mathematics and Its Applications 66. Cambridge University Press, Cambridge, 2008.zbMATHGoogle Scholar
  10. [10]
    D. Cvetković, P. Rowlinson, S. Simić: An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts 75. Cambridge University Press, Cambridge, 2010.Google Scholar
  11. [11]
    M.M. Halldórsson, J. Kratochvíl, J.A. Telle: Independent sets with domination constraints. Discrete Appl. Math. 99 (2000), 39–54.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    W.H. Haemers, M. J.P. Peeters: The maximum order of adjacency matrices of graphs with a given rank. Des. Codes Cryptogr., to appear, doi: 10.1007/s10623-011-9548-3.Google Scholar
  13. [13]
    A. Neumaier: Regular sets and quasi-symmetric 2-designs. Combinatorial theory, Proc. Conf., Schloss Rauischholzhausen 1982, Lect. Notes Math. 969 (1982), 258–275.Google Scholar
  14. [14]
    P. Rowlinson: Star complements in finite graphs: A survey. Rend. Semin. Mat. Messina, Ser. II 8 (2002), 145–162.MathSciNetGoogle Scholar
  15. [15]
    P. Rowlinson: Co-cliques and star complements in extremal strongly regular graphs. Linear Algebra Appl. 421 (2007), 157–162.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    P. Rowlinson: On induced matchings as star complements in regular graphs. J. Math. Sci., New York 182 (2012), 159–163.zbMATHCrossRefGoogle Scholar
  17. [17]
    P. Rowlinson: The main eigenvalues of a graph: a survey. Appl. Anal. Discrete Math. 1 (2007), 445–471.MathSciNetzbMATHGoogle Scholar
  18. [18]
    P. Rowlinson: Regular star complements in strongly regular graphs. Linear Algebra Appl. 436 (2012), 1482–1488.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    J.A. Telle: Characterization of domination-type parameters in graphs. Congr. Numerantium 94 (1993), 9–16.MathSciNetzbMATHGoogle Scholar
  20. [20]
    D.M. Thompson: Eigengraphs: constructing strongly regular graphs with block designs. Util. Math. 20 (1981), 83–115.zbMATHGoogle Scholar
  21. [21]
    F. Zhang: Matrix Theory. Basic Results and Techniques. Universitext. Springer, New York, 1999.CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  • Milica Anđelić
    • 1
    • 2
    Email author
  • Domingos M. Cardoso
    • 1
  • Slobodan K. Simić
    • 3
  1. 1.University of AveiroAveiroPortugal
  2. 2.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia
  3. 3.Mathematical Institute SANUBelgradeSerbia

Personalised recommendations