Geodesic
Bounds and inequalities for the Perron root of a~nonnegative matrix. II.~Circuit bounds and inequalities
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 60-88

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The paper suggests two-sided, upper, and lower circuit bounds for the Perron root of a nonnegative matrix, most of which are derived based on an extension of the monotonicity property of the Perron root established by Fiedler and Pták. 
@article{ZNSL_2003_296_a4,
     author = {L. Yu. Kolotilina},
     title = {Bounds and inequalities for the {Perron} root of a~nonnegative matrix. {II.~Circuit} bounds and inequalities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {60--88},
     publisher = {mathdoc},
     volume = {296},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a4/}
}
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L. Yu. Kolotilina. Bounds and inequalities for the Perron root of a~nonnegative matrix. II.~Circuit bounds and inequalities. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 60-88. http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a4/