Laplacian Eigenvalues of Threshold Graphs

  • Ravindra B. Bapat
Part of the Universitext book series (UTX)


Several problems in mathematics can be viewed as completion problems. Matrix theory is particularly rich in such problems. Such problems nicely blend graph theoretic notions with matrix theory. In this chapter we consider one particular completion problem, the positive definite completion problem, in detail. We first illustrate the concept of matrix completion by considering the nonsingular completion problem. We then introduce the preliminaries on chordal graphs. Then we prove the main result that a graph is positive definite completable if and only if it is chordal.

References and Further Reading

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    Bai, H.: The Grone-Merris conjecture. Trans. Amer. Math. Soc. 363(8), 4463–4474 (2011)Google Scholar
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    Grone, R., Merris, R.: The Laplacian spectrum of a graph II. SIAM J. Discrete Math. 7(2), 221–229 (1994)MathSciNetCrossRefMATHGoogle Scholar
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    Grone, R., Merris, R.: Indecomposable Laplacian integral graphs. Linear Algebra Appl. 428, 1565–1570 (2008)Google Scholar
  4. [MP95]
    Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics, Annals of Discrete Mathematics, 54. North-Holland Publishing Co., Amsterdam (1995)Google Scholar
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    Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering, 143. Academic Press, New York (1979)Google Scholar
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    Merris, R.: Degree maximal graphs are Laplacian integral. Linear Algebra Appl. 199, 381–389 (1994)MathSciNetCrossRefMATHGoogle Scholar
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    So, W.: Rank one perturbation and its application to the Laplacian spectrum of a graph. Linear and Multilinear Algebra 46, 193–198 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia

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