The scrambling index set of primitive minimally strong digraphs

Denote the scrambling index set of primitive minimally strong digraphs of order n   by SInSIn. For n≥13n≥13, we show that SInSIn contains all integers in the interval [3,K1(n)][3,K1(n)], whereK1(n)={n2+12n−138,if both n and n+12 are odd,n2+12n−58,if n is odd and n+12 is even,n2+10n−168,if n is even and n2 is odd,n2+10n−88,if both n and n2 are even. We also identify all elements of SInSIn which are larger than (n2−6n+13)/2(n2−6n+13)/2 (n   is odd) or (n2−6n+12)/2(n2−6n+12)/2 (n   is even). It is shown that there exist more “gaps” in the set SInSIn.