  We study the operator theory associated with such infinite graphs $G$ as
occur in electrical networks, in fractals, in statistical mechanics, and even
in internet search engines. Our emphasis is on the determination of spectral
data for a natural Laplace operator associated with the graph in question. This
operator $\Delta$ will depend not only on $G$, but also on a prescribed
positive real valued function $c$ defined on the edges in $G$. In electrical
network models, this function $c$ will determine a conductance number for each
edge. We show that the corresponding Laplace operator $\Delta$ is automatically
essential selfadjoint. By this we mean that $\Delta$ is defined on the dense
subspace $\mathcal{D}$ (of all the real valued functions on the set of vertices
$G^{0}$ with finite support) in the Hilbert space $l^{2}% (G^{0})$. The
conclusion is that the closure of the operator $\Delta$ is selfadjoint in
$l^{2}(G^{0})$, and so in particular that it has a unique spectral resolution,
determined by a projection valued measure on the Borel subsets of the infinite
half-line. We prove that generically our graph Laplace operator
$\Delta=\Delta_{c}$ will have continuous spectrum. For a given infinite graph
$G$ with conductance function $c$, we set up a system of finite graphs with
periodic boundary conditions such the finite spectra, for an ascending family
of finite graphs, will have the Laplace operator for $G$ as its limit.
