Geodesic
Harary's theorem on signed graphs and reversibility of Markov chains
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVI, Tome 419 (2013), pp. 5-15 
Yu. A. Al'pin. Harary's theorem on signed graphs and reversibility of Markov chains. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVI, Tome 419 (2013), pp. 5-15. http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a0/
@article{ZNSL_2013_419_a0,
     author = {Yu. A. Al'pin},
     title = {Harary's theorem on signed graphs and reversibility of {Markov} chains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--15},
     year = {2013},
     volume = {419},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a0/}
}
TY  - JOUR
AU  - Yu. A. Al'pin
TI  - Harary's theorem on signed graphs and reversibility of Markov chains
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 5
EP  - 15
VL  - 419
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a0/
LA  - ru
ID  - ZNSL_2013_419_a0
ER  - 
%0 Journal Article
%A Yu. A. Al'pin
%T Harary's theorem on signed graphs and reversibility of Markov chains
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 5-15
%V 419
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_419_a0/
%G ru
%F ZNSL_2013_419_a0

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A counterpart of the well-known Harary theorem on signed graphs is proved for digraphs over groups. This result is then used to derive a known theorem on the diagonal similarity of matrices and Kolmogorov's criterion of the reversibility of Markov chains.

[1] A. N. Kolmogoroff, “Zur Theorie der Markoffschen Ketten”, Math. Ann., 112 (1936), 155–160 | DOI | MR

[2] F. P. Kelly, Reversibility and stochastic networks, John Wiley, New York, 1979 | MR | Zbl

[3] F. Harary, “On the notion of balance of a signed graph”, Michigan Math. J., 2 (1954), 143–146 | MR | Zbl

[4] F. S. Roberts, Diskretnye matematicheskie modeli s prilozheniyami k sotsialnym, biologicheskim i ekologicheskim zadacham, Nauka, M., 1986 | MR | Zbl

[5] G. R. Belitskii, Yu. I. Lyubich, Normy matrits i ikh prilozheniya, Naukova dumka, Kiev, 1984 | MR | Zbl

[6] F. Harary, J. K. Kabell, “A simple algorithm to detect balance in signed graphs”, Math. Social Sci., 1 (1980), 131–136 | DOI | MR | Zbl

[7] G. M. Engel, H. Schneider, “Cyclic and diagonal products on a matrix”, Linear Algebra Appl., 7 (1973), 301–335 | DOI | MR | Zbl

[8] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989 | MR