Geodesic
New circuit bounds for the Perron root of a~nonnegative matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 86-103

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The paper suggests new two-sided bounds for the Perron root of a weakly irreducible nonnegative matrix, which depend on the circuits of length no less than two in the associated directed graph and, in some cases, improve Al'pin's bounds, suggested in 1995. Two approaches to deriving circuit bounds are considered. 
@article{ZNSL_2011_395_a8,
     author = {L. Yu. Kolotilina},
     title = {New circuit bounds for the {Perron} root of a~nonnegative matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {86--103},
     publisher = {mathdoc},
     volume = {395},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a8/}
}
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L. Yu. Kolotilina. New circuit bounds for the Perron root of a~nonnegative matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 86-103. http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a8/