  We study upper bounds for the first non-zero eigenvalue of the Steklov
problem defined on finite graphs with boundary. For finite graphs with boundary
included in a Cayley graph associated to a group of polynomial growth, we give
an upper bound for the first non-zero Steklov eigenvalue depending on the
number of vertices of the graph and of its boundary. As a corollary, if the
graph with boundary also satisfies a discrete isoperimetric inequality, we show
that the first non-zero Steklov eigenvalue tends to zero as the number of
vertices of the graph tends to infinity. This extends recent results of Han and
Hua, who obtained a similar result in the case of $\mathbb{Z}^n$. We obtain the
result using metric properties of Cayley graphs associated to groups of
polynomial growth.
