Upper bounds for eigenvalue ratios on general quantum graphs depending only on N and β

Develop, for any compact quantum graph Γ (allowing arbitrary mixes of Dirichlet, Neumann, and Kirchhoff conditions) and any indices k≥j≥1, an upper bound for λk(Γ)/λj(Γ) that depends only on the number N of Neumann leaves and the first Betti number β (in addition to k and j), and does not depend on the number of Dirichlet leaves or other geometric details.

Background

The authors prove a general bound λk/λj ≤ 4·(k2/(j−(N+β))2) valid when j≥N+β+1, showing that cycles and Neumann leaves are the sole obstructions to universal Ashbaugh–Benguria-type bounds on graphs.

They ask for a bound that, for all index pairs (k,j), depends only on N and β (besides indices), thereby removing residual dependencies (e.g., on the number of Dirichlet leaves) and potentially extending validity to j≤N+β cases.

References

Open Problem. Obtain an upper bound on $\frac{\lambda_k(\Gamma)}{\lambda_j(\Gamma)}$, for any compact quantum graph $\Gamma$ and any pair $k \geq j$, where the upper bound should only depend on the number of Neumann leaves $N$ and the number of independent cycles $\beta$ (cf. Theorem~\ref{thm:general-graph-k-j}).

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