  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 [Variational aspects of Laplace eigenvalues on Riemannian
surfaces, Adv. Math. (2014)], 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 $\mathcal{G}$ repsectively.
Secondly, we prove that $\lambda_2 \leq B^2 (D-1)/ |\delta \Omega|$ and
$\lambda_2 \leq (2L + (L-2)(B-2))/L^2$ 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.
