  We give the spectral representation for a class of selfadjoint discrete graph
Laplacians $\Delta$, with $\Delta$ depending on a chosen graph $G$ and a
conductance function $c$ defined on the edges of $G$. We show that the spectral
representations for $\Delta$ fall in two model classes, (1) tree-graphs with
$N$-adic branching laws, and (2) lattice graphs. We show that the spectral
theory of the first class may be computed with the use of rank-one
perturbations of the real part of the unilateral shift, while the second is
analogously built up with the use of the bilateral shift. We further analyze
the effect on spectra of the conductance function $c$: How the spectral
representation of $\Delta$ depends on $c$.
  Using $\Delta_G$, we introduce a resistance metric, and we show that it
embeds isometrically into an energy Hilbert space. We introduce an associated
random walk and we calculate return probabilities, and a path counting number.
