[Math] Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries

Question

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!

Answer 1

For undirected graphs, Theorem 2.2 in this paper might help a bit.

UPDT: Let $G$ be a weighted undirected graph with Laplacian matrix $L$. Let $D$ be a positive diagonal matrix. Let $d=min(diag(D))$ and let $\Delta$ be the maximum diagonal entry of $L$. Let $i$ be the weighted isoperimetric number of $G$. Then: $$ \lambda_{2}(DL) \geq d (\Delta-\sqrt{\Delta^{2}-i^{2}}) $$