This argument essentially depends on the so-called generalized eigenvalues of the Laplacian. In general, given a matrix A, its generalized eigenvalues are described vis-a-vis another matrix B as those values λ for which A - λ B has a non-empty kernel, reducing to the usual definition of eigenvalues if B = I. In the case of the preconditioning argument the matrix A is the Laplacian of the graph we are working with and "preconditioning" it with a subgraph involves working with the generalized eigenvalues, with B being the Laplacian of a subgraph.
For this term paper I would like you to research all you can find about the uses of generalized eigenvalues of graph Laplacians in the specific setting where the second matrix is subgraph of the original graph. Apart from presenting what you find in the literature I also want you to work out interesting examples using simple subgraphs apart from trees, and try to present a picture of what such generalized eigenvalues look like.