Local primitiveness of graphs and nonnegative matrices

Some important properties of objects simulated by nonnegative matrices (graphs) are revealed when their submatrices are positive (subgraphs are complete). For this reason, the primitiveness and the exponent of a matrix (graph) are generalized to the local primitiveness and to the quasiprimitiveness of nonnegative matrices and graphs. Conditions for matrix local primitiveness and quasiprimitiveness are obtained. A relation between local exponent and exponent is established.