Matrices with a transitive graph and inverse M-matrices (Q1084412)

Denote \(M^{-1}\) the class of nonsingular matrices whose inverse is an M-matrix. The relationship between inverse M-matrices and matrices whose graph is transitive is studied. The authors characterize the cases where the ray \(B_{M^{-1}}\{\alpha \in R:\alpha I+B\in M^{-1}\}\) of a nonnegative transitive matrix B is open and closed. These results are applied to obtain a new proof of the characterization due M. Lewin and M. Neumann of (0,1) inverse M-matrices.