  A resistance network is a connected graph $(G,c)$. The conductance function
$c_{xy}$ weights the edges, which are then interpreted as resistors of possibly
varying strengths. The relationship between the natural Dirichlet form
$\mathcal E$ and the discrete Laplace operator $\Delta$ on a finite network is
given by $\mathcal E(u,v) = \la u, \Lap v\ra_2$, where the latter is the usual
$\ell^2$ inner product. We extend this formula to infinite networks, where a
new (boundary) term appears. The Laplace operator is typically unbounded in
this context; we construct a reproducing kernel for the space of functions of
finite energy which allows us to specify a dense domain for $\Delta$ and give
several criteria for the transience of the random walk on the network. The
extended Gauss-Green identity and the reproducing kernel are the foundation for
a boundary integral representation for harmonic functions of finite energy,
akin to that of Martin boundary theory.
