On bounds of extremal eigenvalues of irreducible and m-reducible matrices

We also study bounds for reducible matrices by introducing the notion of m-reducibility. In particular, we give bounds for reducible matrices which depend on the algebraic connectivity of the strongly connected components of the graph and the number of edges between them. These bounds are also applicable to bound the subdominant eigenvalues of reducible stochastic matrices and Laplacian matrices of directed graphs.