Let $\mathcal{G}$ be a strongly connected weighted digraph with corresponding (weighted) adjacency matrix $\mathbf{A}\in\mathbb{R}^{n\times n}_{\geq0}$ and out-degree matrix $\mathbf{D}\in\mathbb{R}^{n\times n}_{>0}$. Moreover, let $\mathbf{L}\in\mathbb{R}^{n\times n}$ be the associated Laplacian matrix, i.e., $\mathbf{L}=\mathbf{D} - \mathbf{A}$.
My questions are the following.
a) I know that the strong connectivity of $\mathcal{G}$ implies that $\mathbf{A}$ is an irreducible matrix. However, what can we say about $\mathbf{L}$? Is $\mathbf{L}$ irreducible as well?
b) Let $\mathbf{B}\in\mathbb{R}^{n\times n}_{\geq0}$ be any diagonal matrix with non-negative diagonal entries and let $\mathbf{M}= \mathbf{L} + \mathbf{B}$. Is $\mathbf{M}$ irreducible?
The goal is to prove that $\mathbf{M}$ is an irreducibly diagonally dominant matrix (which would then imply that $\mathbf{M}$ is nonsingular).
Any formal proof or reference that sheds some light onto my questions would be greatly appreciated. The main reference that I have been using so far is Matrix Analysis, Horn and Johnson, 2nd Ed.