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 $\delta \Omega$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $\lambda_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, \delta \Omega)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $\lambda_2 \leq 8D / |\delta \Omega|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound, that is, $\lambda_2 < 8\pi / |\partial \Omega|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ respectively. Secondly, we prove that $\lambda_2 \leq 4 (B-1) (D-1)/ |\delta \Omega|$ and $\lambda_2 \leq B/L$ for block graphs, which extend the results on trees by He and Hua. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given.