A Class of M -Matrices with Tree Graphs
Charles R. Johnson, Dale D. Olesky, Pauline Van den Driessche · SIAM Journal on Algebraic and Discrete Methods · 1983
A class of matrices is considered for which the directed graphs have a longest simple circuit of length two; for an irreducible matrix this means that its undirected graph is a tree. A matrix in this class which is both positive stable and inverse nonnegative is proved to be an M-matrix. A characterization is given of those inverse M-matrices for which the corresponding M-matrix lies in this class. The results are related to known theorems on tridiagonal matrices.