## Abstract

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 language | English |
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Pages (from-to) | 2143-2168 |

Number of pages | 26 |

Journal | Linear and Multilinear Algebra |

Volume | 69 |

Issue number | 11 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2019 Informa UK Limited, trading as Taylor & Francis Group.

### Funding

The work is supported by the grant Russian Science Foundation (RSF) 17-11-01124. The authors are grateful to the referee for valuable comments.

Funders | Funder number |
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Russian Science Foundation | 17-11-01124 |

## Keywords

- Graphs
- primitive matrices
- scrambling index