Eigenvalue estimates for the Laplacian on a metric tree
Spectral Theory
2016-07-28 v3 Mathematical Physics
math.MP
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.
Cite
@article{arxiv.1602.03864,
title = {Eigenvalue estimates for the Laplacian on a metric tree},
author = {Jonathan Rohleder},
journal= {arXiv preprint arXiv:1602.03864},
year = {2016}
}
Comments
revised version; accepted for publication in Proc. Amer. Math. Soc