Let $A \in \mathbb{R}_+^{n\times n}$ be the adjacency matrix of a weighted directed graph, i.e., $A$ is nonsymmetric and with nonnegative entries. Let $M = D^{-1}A$ be the row-stochastic Markov matrix associated with the graph, hence $D$ is the diagonal matrix whose elements are the row-sums of $A$.
I know that from the Perron-Frobenius theorem the eigenvalues $\lambda$ of $M$ are contained within a unit disk centered in 0, and that particularly $\lambda_1 = 1$ where $\lambda_1 > |\lambda_2| > \cdots > |\lambda_n|$.
I was wondering if there were any conditions on $A$ that allow to bound the eigenvalues s.t. their real part is nonnegative, i.e., $\Re(\lambda_n) \geq 0$.
One sufficient, but not necessary condition is that the symmetric part of $A$ is positive semidefinite, i.e., $\lambda(A+A^T) \geq 0$. Are there other conditions of this type and/or sufficient and necessary conditions that can be stated in terms e.g. of the eigenvalues of $A$ or its entries?