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 language | English |
|---|---|
| Pages (from-to) | 691-707 |
| Number of pages | 17 |
| Journal | Kybernetika |
| Volume | 58 |
| Issue number | 5 |
| DOIs | |
| State | Published - 2022 |
| Externally published | Yes |
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.
| Funders | Funder number |
|---|---|
| GENERALIZATION | 101039692 |
| European Commission | |
| European Commission |
Keywords
- cyclicity index
- linear transformations
- tropical linear algebra