Geodesic
Combinatorial and spectral properties of semigroups of stochastic matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVIII, Tome 439 (2015), pp. 13-25 
Yu. A. Al'pin; V. S. Al'pina. Combinatorial and spectral properties of semigroups of stochastic matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVIII, Tome 439 (2015), pp. 13-25. http://geodesic.mathdoc.fr/item/ZNSL_2015_439_a1/
@article{ZNSL_2015_439_a1,
     author = {Yu. A. Al'pin and V. S. Al'pina},
     title = {Combinatorial and spectral properties of semigroups of stochastic matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {13--25},
     year = {2015},
     volume = {439},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_439_a1/}
}
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The paper studies the notion of imprimitivity index of a semigroup of nonnegative matrices, introduced by Protasov and Voynov. A new characterization of the imprimitivity index in terms of the scrambling rank of a nonnegative matrix is suggested. Based on this characterization, an independent combinatorial proof of the Protasov–Voynov theorem on the interrelation between the imprimitivity index of a semigroup of stohastic matrices and the spectral properties of matrices in the semigroup is presented.