Linear Transformations Preserving the Minimal Values of the Cyclicity Index of Tropical Matrices
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 cyc...
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| Published in: | Journal of mathematical sciences (New York, N.Y.) Vol. 281; no. 2; pp. 234 - 245 |
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| Abstract | 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 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 only one value of the cyclicity index. |
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| AbstractList | 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 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 only one value of the cyclicity index. |
| Author | Vlasov, A. V. Kreines, E. M. Guterman, A. E. |
| Author_xml | – sequence: 1 givenname: A. E. surname: Guterman fullname: Guterman, A. E. email: alexander.guterman@biu.ac.il organization: Bar-Ilan University, Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics – sequence: 2 givenname: E. M. surname: Kreines fullname: Kreines, E. M. organization: Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics, Tel Aviv University – sequence: 3 givenname: A. V. surname: Vlasov fullname: Vlasov, A. V. organization: Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics |
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| References | Kennedy-Cochran-PatrickAMerletGNowakTSergeevSNew bounds on the periodicity transient of the powers of a tropical matrix: Using cyclicity and factor rankLinear Algebra Appl.2021611279309419062510.1016/j.laa.2020.10.032 LiC-KTsingNKLinear preserver problems: A brief introduction and some special techniquesLinear Algebra Appl.1992162–164217235114840110.1016/0024-3795(92)90377-M GutermanAKreinesEVlasovANon-surjective linear transformations of tropical matrices preserving the cyclicity indexKybernetika20225856917074538621 G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitz. Deutsch. Akad. Wiss., Berlin (1897). GavalecMLinear matrix period in max-plus algebraLinear Algebra Appl.2000307167182174192410.1016/S0024-3795(00)00020-3 S. Pierce and others, “A survey of linear preserver problems,” Linear Multilinear Algebra, 33, 1–119 (1992). M. Gavalec, Periodicity in Extremal Algebras, Gaudeamus, Hradec Kr´alov´e (2004). GutermanAKreinesEThomassenCLinear transformations of tropical matrices preserving the cyclicity indexSpecial Matrices20219112118422386610.1515/spma-2020-0128 B. Heidergott, G. J. Olsder, and J. van der Woude, Max Plus at Work, Princeton Series in Applied Mathematics (2006). L. Molnar, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces (Lect. Notes Math., 1895) (2007). SergeevSMax algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classesLinear Algebra Appl.200943113251339254791410.1016/j.laa.2009.04.027 7096_CR9 S Sergeev (7096_CR11) 2009; 431 C-K Li (7096_CR8) 1992; 162–164 A Kennedy-Cochran-Patrick (7096_CR7) 2021; 611 A Guterman (7096_CR4) 2021; 9 7096_CR10 A Guterman (7096_CR5) 2022; 58 7096_CR6 7096_CR1 M Gavalec (7096_CR3) 2000; 307 7096_CR2 |
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| SubjectTerms | Graph theory Linear transformations Mathematical analysis Mathematics Mathematics and Statistics Matrices (mathematics) |
| Title | Linear Transformations Preserving the Minimal Values of the Cyclicity Index of Tropical Matrices |
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