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 - 1 Jan 2022 |
Keywords
- cyclicity index
- linear transformations
- tropical linear algebra
All Science Journal Classification (ASJC) codes
- Software
- Control and Systems Engineering
- Theoretical Computer Science
- Information Systems
- Artificial Intelligence
- Electrical and Electronic Engineering