Preserving λ-scrambling matrices

Alexander E. Gutermany, Artem M. Maksaev

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


The notion of scrambling index was firstly introduced by Akelbek and Kirkland in 2009. For a primitive digraph D, it is defined as the smallest positive integer k such that for every pair of vertices u and v of D there exist two directed paths of lengths k to a common vertex w. This notion turned out to be useful for several applications, e. g., to estimate eigenvalues of non-negative primitive stochastic matrices. In 2010 Huang and Liu with the background of a memoryless communication system generalized this notion to λ-tuples of vertices and named it λ-th upper scrambling index. These notions can be reformulated in terms of matrix theory. A standard way to generate matrices with the given λ-th upper scrambling index is to apply certain matrix transformations that preserve this index to the existing examples of matrices with known λ-th upper scrambling index. In this paper we completely characterize bijective linear maps preserving λ-th upper scrambling index 1 or 0.

Original languageEnglish
Pages (from-to)119-141
Number of pages23
JournalFundamenta Informaticae
Issue number2-3
StatePublished - 2018
Externally publishedYes

Bibliographical note

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  • Directed graphs
  • Nonnegative matrices
  • Scrambling index
  • Scrambling matrix


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