The first Steklov eigenvalue of planar graphs and beyond

The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let D and δ Ω be the maximum vertex degree and the set of vertices of degree one in a graph G respectively. Let λ₂ be the first (non-trivial) Steklov eigenvalue of (G, δ Ω). In this paper, using the circle packing theorem and conformal mapping, we first show that λ₂ ≤ 8D / |δ Ω| for planar graphs. This can be seen as a discrete analogue of Kokarev's bound [Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math. (2014)], that is, λ₂ < 8π / |∂ Ω| for compact surfaces with boundary of genus $0$. Let B and L be the maximum block size and the diameter of a block graph G repsectively. Secondly, we prove that λ₂ ≤ B² (D-1)/ |δ Ω| and λ₂ ≤ (2L + (L-2)(B-2))/L² for block graphs, which extend the results on trees by He and Hua [Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations (2022)]. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given.