Equality characterization in the Pólya-type lower bound for Dirichlet trees

Characterize all compact Dirichlet trees Γ of total length L and integers k≥1 for which equality holds in the Pólya-type lower bound λk(Γ) ≥ (π2 k2)/L2; specifically, determine whether equality holds if and only if every edge length is an integer multiple of L/k.

Background

The paper notes that the inequality λk(Γ) ≥ (π2 k2)/L2 holds for all Dirichlet trees and all k, a special case of a more general result. However, the equality case is not addressed in prior work or here.

The authors propose a natural structural conjecture for equality, motivated by eigenfunction structures on equilateral and commensurate graphs, which would provide a precise geometric characterization of extremizers.

References

Open Problem. Characterize the case of equality in the P\n olya-type bound eq:dirichlet-tree-polya, $\lambda_k(\Gamma) \geq \frac{\pi2 k2}{L2}$, for all $k \geq 1$ and all Dirichlet trees $\Gamma$ of total length $L$. Is it true that there is equality if and only if all edges of $\Gamma$ have length equal to an integer multiple of $\frac{L}{k}$?

— Bounds on eigenvalue ratios of quantum graphs  (2603.26172 - Harrell et al., 27 Mar 2026) in Open Problem, end of Introduction (Section 1)