Maximizing the Steklov eigenvalues on trees with a diameter constraint

TL;DR

This paper characterizes the extremal trees maximizing the first nonzero Steklov eigenvalue under diameter constraints, completing the geometric classification for all diameters.

Contribution

It determines the precise structure of trees achieving the maximum Steklov eigenvalue for all odd diameters, extending previous results for even diameters and diameter three.

Findings

Maximum eigenvalue achieved on generalized almost seesaw trees

Complete classification for all diameters

Introduces a boundary variational approach for trees

Abstract

We study the first nonzero Steklov eigenvalue $\lambda_2(T,\delta\Omega)$ of the Dirichlet-to-Neumann operator on a finite tree $T$ with leaf boundary $\delta\Omega$, under a constraint on the diameter $D$. He and Hua [Calc. Var. PDE, 2022] showed that $\lambda_2(T) \leq 2/D$ for any tree of diameter $D$, with the even-diameter equality case fully characterized. For odd $D$, the geometric picture underlying the sharp configurations has remained unclear beyond diameter three. We determine this picture completely for all odd diameters $D = 2r+1 \geq 5$. The sharp value of $\lambda_2$ is achieved on spider trees with nearly-equidistributed branch lengths, forming the family of \emph{generalized almost seesaw trees} $\mathrm{AS}(r,q+2,c,t)$, prescribed by the arithmetic of $n$ relative to $\lceil r/2 \rceil$. Together with the results of He-Hua and Lin-Zhao [Bull. Lond. Math. Soc., 2025] for even diameters and diameter three, this completes the geometric classification for every diameter. The argument is based on a scalar root equation for one-center profiles, an inverse boundary quadratic form on boundary fluxes, and a reduction scheme from arbitrary trees to two-center profiles, and then to the one-center class. The inverse variational viewpoint may be regarded as a boundary analogue of the classical distance-matrix formalism for trees initiated by Graham and Lov\'asz [Adv. Math., 1978].