Geodesic
Linear transformations preserving minimal values of the cyclicity index of tropical matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVI, Tome 524 (2023), pp. 18-35 
A. V. Vlasov; A. E. Guterman; E. M. Kreines. Linear transformations preserving minimal values of the cyclicity index of tropical matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVI, Tome 524 (2023), pp. 18-35. http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a2/
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     author = {A. V. Vlasov and A. E. Guterman and E. M. Kreines},
     title = {Linear transformations preserving minimal values of the cyclicity index of tropical matrices},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a2/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The cyclicity index of a directed graph is defined as the least common multiple of the cyclicity indices of all its strongly connected components, and the cyclicity index of a strongly connected directed graph is equal to the greatest common divisor of the lengths of all its directed cycles. The cyclicity index of a tropical matrix is the cyclicity index of its critical subgraph, i.e., the subgraph of the adjacency graph, consisting of all cycles with the largest average weight. This paper considers linear transformations of tropical matrices that preserve only two values of the cyclicity index, 1 and 2. A complete characterization of such transformations is obtained. To this end, it is proved that the values 1 and 2 of the cyclicity index are preserved if and only if all its values are preserved. It is shown that there are mappings of another type that preserve one fixed value of the cyclicity index.

[1] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitz. Deutsch. Akad. Wiss., Berlin, 1897

[2] M. Gavalec, Periodicity in Extremal Algebras, Gaudeamus, Hradec Králové, 2004

[3] M. Gavalec, “Linear matrix period in max-plus algebra”, Linear Algebra Appl., 307 (2000), 167–182 | DOI | MR | Zbl

[4] A. Guterman, E. Kreines, C. Thomassen, “Linear transformations of tropical matrices preserving the cyclicity index”, Special Matrices, 9 (2021), 112–118 | DOI | MR | Zbl

[5] A. Guterman, E Kreines, A. Vlasov, “Non-surjective linear transformations of tropical matrices preserving the cyclicity index”, Kybernetika, 58:5 (2022), 691–707 | MR

[6] B. Heidergott, G. J. Olsder, J. van der Woude, Max Plus at Work, Princeton Series in Applied Mathematics, 2006 | MR | Zbl

[7] A. Kennedy-Cochran-Patrick, G. Merlet, T. Nowak, S. Sergeev, “New bounds on the periodicity transient of the powers of a tropical matrix: Using cyclicity and factor rank”, Linear Algebra Appl., 611 (2021), 279–309 | DOI | MR | Zbl

[8] C.-K. Li, N. K. Tsing, “Linear preserver problems: A brief introduction and some special techniques”, Linear Algebra Appl., 162–164 (1992), 217–235 | MR | Zbl

[9] L. Molnar, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lect. Notes Math., 1895, 2007 | MR | Zbl

[10] S. Pierce and others, “A survey of linear preserver problems”, Linear Multilinear Algebra, 33 (1992), 1–119 | DOI | MR | Zbl

[11] S. Sergeev, “Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes”, Linear Algebra Appl., 431 (2009), 1325–1339 | DOI | MR | Zbl