I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since the graph is strongly connected, $L$ has a simple zero eigenvalue and all its nonzero eigenvalues have positive real part. Is it possible to establish a relation e.g., a bound, between the eigenvalues of $L$ and those of the product $DL$?

Thanks a lot!