Does there exist a universal constant C>0 such that, for every connected graph G=(V,E) with boundary B, maximum degree d_max, and genus g, one has sigma_k(G,B)≤ C d_maxfracg+k|B| for every integer k satisfying 1≤ k≤ |B|, where sigma_k(G,B) is the k-th Steklov eigenvalue of G with boundary B?

Lin–Zhao graph Steklov eigenvalue problem

The problem asks for upper bounds on graph Steklov eigenvalues in terms of genus, maximum degree, boundary size, and eigenvalue index. Lin and Zhao posed the positive-genus extension, now claimed for $1\le k\le |B|$ by a universal estimate.

Known results

Planar graphs admit bounds of order $Dk/|B|$ for higher Steklov eigenvalues.
Earlier positive-genus bounds were $O\!\left(Dg(\log g)^2k/|B|\right)$ , with removal of the logarithmic factor identified as open.
A prior first-eigenvalue estimate had order $\Delta(g+1)^3/|B|$ under additional size assumptions.
A newer partial result gives an $O\!\left(g/|B|\right)$ bound for the first nontrivial eigenvalue in positive genus.

August 2026 transfer-principle claim

An August 18, 2026 arXiv preprint claims a graph-to-surface transfer principle proving $$\sigma_k(G,B)\le C d_{\max}\frac{g+k}{|B|}. $It calls the estimate sharp up to a universal constant and says this completely resolves the problem; taking$ B=V$ yields the corresponding Laplacian bound.

Current status (as of August 2026): The full estimate is claimed in an arXiv preprint, but independent verification and peer review are absent; no counterexample or reported proof gap is recorded.

Lin–Zhao graph Steklov eigenvalue problem

Progress summary

Primary source

Additional references

Solutions 0