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A Conjecture on Laplacian Eigenvalues of Trees

  • David P. Jacobs
  • Vilmar TrevisanEmail author
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

Motivated by classic tree algorithms, in 1995 we designed a bottom-up O(n) algorithm to compute the determinant of a tree’s adjacency matrix A. In 2010 an O(n) algorithm was found for constructing a diagonal matrix congruent to A + xIn, \(x \in \mathbb {R}\), enabling one to easily count the number of eigenvalues in any interval. A variation of the algorithm allows Laplacian eigenvalues in trees to be counted. We conjecture that for any tree T of order n ≥ 2, at least half of its Laplacian eigenvalues are less than \(\bar {d} = 2 - \frac {2}{n} \), its average vertex degree.

Notes

Acknowledgements

The authors are grateful to Science without Borders CNPq grant 400122/2014-6 which provided support for this work. V. Trevisan also ackowledges the support from CAPES under grant MATHAmSud 88881.143281/2017-01 and FAPERGS under grant PqG 17/2551-01.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Clemson UniversityClemsonClemsonUSA
  2. 2.UFRGSPorto AlegreBrazil

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