Eigenvalue estimates for the Laplacian on a metric tree

Rohleder, Jonathan

doi:10.1090/proc/13403

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
MathSciNet review: 3611325
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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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