Sufficient conditions for local primitiveness of nonprimitive digraphs

For some communication systems modelled by nonnegative matrices, some important properties are reached if certain submatrices of the matrix degree are positive. In order to investigate these properties, the concepts of local primitiveness and local exponent of the matrix (digraph) connected with positivity of a certain submatrix (subgraph) of this matrix are introduced. Several sufficient conditions of local primitiveness and some bounds of local exponents for nonprimitive digraphs are presented.