Geodesic
Non-surjective linear transformations of tropical matrices preserving the cyclicity index
Kybernetika, Tome 58 (2022) no. 5, pp. 691-707

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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.
DOI : 10.14736/kyb-2022-5-0691
Classification : 05C22, 05C38, 05C50
Keywords: tropical linear algebra; cyclicity index; linear transformations 
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Guterman, Alexander; Kreines, Elena; Vlasov, Alexander. Non-surjective linear transformations of tropical matrices preserving the cyclicity index. Kybernetika, Tome 58 (2022) no. 5, pp. 691-707. doi: 10.14736/kyb-2022-5-0691

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