spectral graph theory with "potentials"

Question

Let G be an undirected graph with bounded degree and n vertices. Let L[G] be the corresponding graph Laplacian, which is a symmetric $n \times n$ matrix. Let V be an $n \times n$ diagonal matrix. I am interested in estimating the gap between the lowest and second lowest eigenvalues of matrices of the form -L[G]+V for various graphs G and diagonal matrices V. I am aware that a lot is known about the gap between the lowest and second lowest eigenvalues of -L[G]. My understanding is that this is the main content of the subject called spectral graph theory. However, I have not found many references on the more general problem, where we add a diagonal term (which I think of as being analogous to the potential term in Schroedinger's equation). I would be most grateful for any suggestions of relevant references or keywords.