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In this paper, some estimations for the spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices are given by matrix directed graphs and their k-path covering. The existent results on the upper and lower bounds of the spectral radius of nonnegative matrices are improved. MSC: 15A18; 65F15doi:10.1186/1029-242x-2014-477 fatcat:fjqr3g34ubcbvfsg7cuake2u4y