Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc