The scrambling index of primitive digraphs

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph DD, which is the smallest positive integer kk such that for every pair of vertices uu and vv, there is a vertex ww such that we can get to ww from uu and vv in DD by directed walks of length kk. In this paper, we obtain some new upper bounds for the scrambling index of primitive digraphs. Moreover, the maximum index problem, the extremal matrix problem and the index set problem for the scrambling index of various classes of primitive digraphs (e.g. primitive digraphs with dd loops, minimally strong digraphs, nearly decomposable digraphs, micro-symmetric digraphs, etc.) are settled, respectively.