Existence of maximizers for λk/λj with only N and β constrained

Ascertain, for each fixed N≥0 and β≥0, whether there exists (for every pair of indices k≥j≥1) a compact quantum graph with at most N Neumann leaves and β independent cycles that maximizes λk(Γ)/λj(Γ) without imposing any restriction on the number of Dirichlet leaves.

Background

The paper proves existence of maximizers (and minimizers) for λk/λj within classes of graphs with a bounded number of edges, and, more generally, with bounded counts of Dirichlet and Neumann leaves and cycles.

They highlight as a major open issue whether the edge-count (or Dirichlet-leaf) restrictions can be removed, leaving only constraints on N and β while still guaranteeing the existence of maximizers.

References

Open Problem. Study whether, for any given $N \geq 0$ and $\beta \geq 0$, among all compact graphs with at most $N$ Neumann leaves and $\beta$ independent cycles, for any $k \geq j \geq 1$ there exists a graph maximizing $\frac{\lambda_k}{\lambda_j}$ (independent of the number of Dirchlet leaves) cf. Corollary~\ref{cor:existence}.

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