NON-SURJECTIVE LINEAR TRANSFORMATIONS OF TROPICAL MATRICES PRESERVING THE CYCLICITY INDEX

Alexander Guterman, Elena Kreines, Alexander Vlasov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The cyclicity index of a matrix is the cyclicity index of its critical subgraph, namely, the subgraph of the adjacency graph which consists of all cycles of the maximal average weight. The cyclicity index of a graph is the least common multiple of the cyclicity indices of all its maximal strongly connected subgraphs, and the cyclicity index of a strongly connected graph is the least common divisor of the lengths of its (directed) cycles. In this paper we obtain the characterization of linear, possibly non-surjective, transformations of tropical matrices preserving the cyclicity index. It appears that non-bijective maps with these properties exist and all maps are exhausted by transposition, renumbering of vertices, Hadamard multiplication with a matrix of a certain special structure, and certain diagonal transformation. Moreover, only diagonal transformation can be non-bijective.

Original languageEnglish
Pages (from-to)691-707
Number of pages17
JournalKybernetika
Volume58
Issue number5
DOIs
StatePublished - 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022 Institute of Information Theory and Automation of The Czech Academy of Sciences. All rights reserved.

Funding

The authors are grateful to the referees for careful reading and valuable suggestions. The first and the second authors wish to thank Technion – Israel Institute of Technology for the warm hospitality. The work of the first author was funded by the European Union (ERC, GENERALIZATION, 101039692). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

FundersFunder number
GENERALIZATION101039692
European Commission
European Research Council

    Keywords

    • cyclicity index
    • linear transformations
    • tropical linear algebra

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