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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Eigenvalue estimates for the Laplacian on a metric tree
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by Jonathan Rohleder PDF
Proc. Amer. Math. Soc. 145 (2017), 2119-2129 Request permission

Abstract:

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e., the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.
References
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Additional Information
  • Jonathan Rohleder
  • Affiliation: TU Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, Gebäude E, 21073 Hamburg, Germany
  • MR Author ID: 980414
  • Email: jonathan.rohleder@tuhh.de
  • Received by editor(s): February 17, 2016
  • Received by editor(s) in revised form: July 4, 2016
  • Published electronically: October 27, 2016
  • Additional Notes: The author wishes to thank Gregory Berkolaiko for drawing his attention to the eigenvalue inequalities contained in \cite{BBW15}. Moreover, the author gratefully acknowledges financial support from the Austrian Science Fund (FWF), project P 25162-N26.
  • Communicated by: Michael Hitrik
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2119-2129
  • MSC (2010): Primary 34B45; Secondary 81Q10, 81Q35
  • DOI: https://doi.org/10.1090/proc/13403
  • MathSciNet review: 3611325