Infinite networks and variation of conductance functions in discrete Laplacians

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 $c$.