Geodesic
An upper bound for the chainable index
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 5-17

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The paper considers the chainable index of a square matrix of order $n$ and proves that it does not exceed $n-1$. Also it is demonstrated that every integer in between $0$ and $n-1$ is a value of the chainable index. 
@article{ZNSL_2022_514_a0,
     author = {Yu. A. Alpin and A. E. Guterman and E. R. Shafeev},
     title = {An upper bound for the chainable index},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--17},
     publisher = {mathdoc},
     volume = {514},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_514_a0/}
}
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Yu. A. Alpin; A. E. Guterman; E. R. Shafeev. An upper bound for the chainable index. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 5-17. http://geodesic.mathdoc.fr/item/ZNSL_2022_514_a0/