Upper bounds on scrambling index for non-primitive digraphs

A. E. Guterman, A. M. Maksaev

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)2143-2168
Number of pages26
JournalLinear and Multilinear Algebra
Issue number11
StatePublished - 2021
Externally publishedYes

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  • Graphs
  • primitive matrices
  • scrambling index


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