Quantum trees which maximize higher eigenvalues are unbalanced

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th positive eigenvalue is the star graph with three edges of lengths $2 k - 1$, $1$ and $1$. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape -- an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.

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