Consider an connected digraph, we use the classic definition of the Laplacian matrix $L$:
$L=D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix.
There has been many researches on how the eigenvalue will evolve when adding edges to the graph, (such as Bojan Mohar, the interlace of the eigenvalues, etc.).
But I am wondering if there is some work focus on the eigenvalue ratio of the Laplacian matrix, for instance, $\frac{\lambda_i}{\lambda_2}$, where $0=\lambda_1<\lambda_2\leq\cdots\leq\lambda_N$ are the eigenvalues of the Laplacian.