Geodesic
Bounds and inequalities for the Perron root of a nonnegative matrix. III.~Bounds dependent on simple paths and circuits
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 69-93

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The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding $k$ and the simple paths of length $k$, $1\le k\le n$, in the directed graph of the matrix. For each $k$, $1\le k\le n$, these bounds are intermediate between the circuit bounds and the path-dependent bounds suggested previously, and for $k=1$ and $k=n$ they reduce to the corresponding path-dependent bounds and the circuit bounds, respectively. 
@article{ZNSL_2005_323_a7,
     author = {L. Yu. Kolotilina},
     title = {Bounds and inequalities for the {Perron} root of a nonnegative matrix. {III.~Bounds} dependent on simple paths and circuits},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {69--93},
     publisher = {mathdoc},
     volume = {323},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a7/}
}
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L. Yu. Kolotilina. Bounds and inequalities for the Perron root of a nonnegative matrix. III.~Bounds dependent on simple paths and circuits. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 69-93. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a7/