## Abstract

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 language | English |
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Pages (from-to) | 119-141 |

Number of pages | 23 |

Journal | Fundamenta Informaticae |

Volume | 162 |

Issue number | 2-3 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018 IOS Press. All rights reserved.

## Keywords

- Directed graphs
- Nonnegative matrices
- Scrambling index
- Scrambling matrix