We study Laplace operators on infinite networks $(G,c)$, and their
self-adjoint extensions. We consider the Laplacian $\Delta$ as on operator on
$\ell^2(G)$ and as an operator on the Hilbert space $\mathcal{H}_\mathcal{E}$
of finite energy functions on $G$, focusing on the case when $\Delta$ is
unbounded. It is known that $\Delta$ is essentially self-adjoint on its natural
domain in $\ell^2(G)$, but that it is \emph{not} essentially self-adjoint on
its natural domain in $\mathcal{H}_\mathcal{E}$.

In this paper, we consider certain elements in von Neumann algebras generated
by graph groupoids. In particular, we are interested in finitely supported
elements, called graph operators. We study the characterizations for
self-adjointness, the unitary property, hyponormality and normality of graph
operators.

A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance)
metric $R$. We embed the resistance network into the Hilbert space ${\mathcal
H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to
study ${\mathcal H}_{\mathcal E}$, and vice versa and show that the embedded
images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert
space.

A resistance network is a connected graph $(G,c)$. The conductance function
$c_{xy}$ weights the edges, which are then interpreted as conductors of
possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a
Hilbert space structure ${\mathcal H}_{\mathcal E}$ on the space of functions
of finite energy.