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

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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. 
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     author = {Yu. A. Al'pin},
     title = {Harary's theorem on signed graphs and reversibility of {Markov} chains},
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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/