  For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed
conductance function $c$, we study an associated graph Laplacian $\Delta_{c}$;
it is a generalized difference operator where the differences are measured
across the edges $E$ in $G$; and the conductance function $c$ represents the
corresponding coefficients. The graph Laplacian (a key tool in the study of
infinite networks) acts in an energy Hilbert space $\mathscr{H}_{E}$ computed
from $c$. Using a certain Parseval frame, we study the spectral theoretic
properties of graph Laplacians. In fact, for fixed $c$, there are two versions
of the graph Laplacian, one defined naturally in the $l^{2}$ space of $V$, and
the other in $\mathscr{H}_{E}$. The first is automatically selfadjoint, but the
second involves a Krein extension. We prove that, as sets, the two spectra are
the same, aside from the point 0. The point zero may be in the spectrum of the
second, but not the first.
  We further study the fine structure of the respective spectra as the
conductance function varies; showing now how the spectrum changes subject to
variations in the function $c$.
