Advertisement

Graphs and Combinatorics

, Volume 31, Issue 1, pp 59–72 | Cite as

Spectral Characterization of Families of Split Graphs

  • Milica Anđelić
  • Domingos M. CardosoEmail author
Original Paper

Abstract

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted.

Keywords

Split graph Largest eigenvalue Principal eigenvector Programming involving graphs 

Mathematics Subject Classification (2000)

05C50 90C35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cardoso D.M., Kamiński M., Lozin V.V.: Maximum k-regular induced subgraphs. J. Comb. Optim. 14, 455–463 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Cardoso D.M., Rowlinson P.: Spectral upper bounds for the order of a k-regular induced subgraph. Linear Algebra Appl. 433, 1031–1037 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cioabă S.M.: A necessary and sufficient eigenvector condition for a connected graph to be bipartite. Electron. J. Linear Algebra. 20, 351–353 (2010)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cvetković D.M.: Inequalities obtained on the basis of the spectrum of the graph. Studia Sci. Math. Hung. 8, 433–436 (1973)Google Scholar
  5. 5.
    Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs, Theory and Applications, 3rd edn. Johan Ambrosius Barth Verlag, Heidelberg (1995)Google Scholar
  6. 6.
    Cvetković, D.M., Rowlinson, P., Simić, S.K.: An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts. Cambridge University Press, Cambridge (2010)Google Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

Personalised recommendations