Geodesic
The generalized monotonicity property of the Perron root
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 13-29

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The paper presents a new monotonicity property of the Perron root of a nonnegative matrix. It is shown that this new property implies known monotonicity properties and also the Chistyakov two-sided bounds for the Peroon root of a block-partitioned nonnegative matrix. Moreover, based on the monotonicity property suggested, the equality cases in Chistyakov's theorem are analyzed. Applications to bounding above the spectral radius of a complex matrix are presented. 
@article{ZNSL_2006_334_a1,
     author = {Yu. A. Alpin and L. Yu. Kolotilina},
     title = {The generalized monotonicity property of the {Perron} root},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {13--29},
     publisher = {mathdoc},
     volume = {334},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a1/}
}
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Yu. A. Alpin; L. Yu. Kolotilina. The generalized monotonicity property of the Perron root. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 13-29. http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a1/