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Laplacian Eigenvalues of Threshold Graphs

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Graphs and Matrices

Part of the book series: Universitext ((UTX))

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Abstract

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.

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References and Further Reading

  1. Bai, H.: The Grone-Merris conjecture. Trans. Amer. Math. Soc. 363(8), 4463–4474 (2011)

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  2. Grone, R., Merris, R.: The Laplacian spectrum of a graph II. SIAM J. Discrete Math. 7(2), 221–229 (1994)

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  3. Grone, R., Merris, R.: Indecomposable Laplacian integral graphs. Linear Algebra Appl. 428, 1565–1570 (2008)

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  4. Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics, Annals of Discrete Mathematics, 54. North-Holland Publishing Co., Amsterdam (1995)

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  5. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering, 143. Academic Press, New York (1979)

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  6. Merris, R.: Degree maximal graphs are Laplacian integral. Linear Algebra Appl. 199, 381–389 (1994)

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  7. So, W.: Rank one perturbation and its application to the Laplacian spectrum of a graph. Linear and Multilinear Algebra 46, 193–198 (1999)

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Authors and Affiliations

  1. Indian Statistical Institute, New Delhi, India

    Ravindra B. Bapat

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  1. Ravindra B. Bapat
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Correspondence to Ravindra B. Bapat .

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Bapat, R.B. (2014). Laplacian Eigenvalues of Threshold Graphs. In: Graphs and Matrices. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6569-9_11

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