Eigenvalue estimates for the Laplacian on a metric tree

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.

• Publication year

  2016

• Venue

  arXiv: Spectral Theory

• Publication date

  2016-02-11

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1090/PROC/13403arXiv 1602.03864
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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