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Eigenvalue estimates for the Laplacian on a metric tree


Author: Jonathan Rohleder
Journal: Proc. Amer. Math. Soc. 145 (2017), 2119-2129
MSC (2010): Primary 34B45; Secondary 81Q10, 81Q35
DOI: https://doi.org/10.1090/proc/13403
Published electronically: October 27, 2016
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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.


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Additional Information

Jonathan Rohleder
Affiliation: TU Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, Gebäude E, 21073 Hamburg, Germany
Email: jonathan.rohleder@tuhh.de

DOI: https://doi.org/10.1090/proc/13403
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 [1]. Moreover, the author gratefully acknowledges financial support from the Austrian Science Fund (FWF), project P 25162-N26.
Communicated by: Michael Hitrik
Article copyright: © Copyright 2016 American Mathematical Society