Upper bounds on scrambling index for non-primitive digraphs

The notion of the scrambling index is a fundamental invariant in graph theory and in the theory of non-negative matrices and their applications. Namely, a scrambling index of a primitive directed graph G is the smallest positive integer (Formula presented.) such that for any pair of vertices u,v of G there exists a vertex w of G such that there are directed walks of length k from u to w and from v to w. In this paper, we generalize the definition to arbitrary directed graphs. We describe constructively the class of graphs with non-zero scrambling index and generalize the Akelbek–Kirkland bounds for the scrambling index to arbitrary directed graphs. Also, the directed graphs with extremal scrambling index are characterized.