Bounding the gap between extremal Laplacian eigenvalues of graphs

Let G be a graph whose Laplacian eigenvalues are 0 = λ1 ⩽ λ2 ⩽ ⋯ ⩽ λn. We investigate the gap (expressed either as a difference or as a ratio) between the extremal non-trivial Laplacian eigenvalues of a connected graph (that is λn and λ2). This gap is closely related to the average density of cuts in a graph. We focus here on the problem of bounding the gap from below.