Quantum trees which maximize higher eigenvalues are unbalanced

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