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Regular stochastic matrices and digraphs

Published online by Cambridge University Press:  14 July 2016

Elizabeth Berman
Elizabeth Berman*
Affiliation:
Rockhurst College, Kansas City, Missouri
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Abstract

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.

Keywords

REGULAR STOCHASTIC MATRICESERGODIC MARKOV CHAINSSTRONGLY CONNECTED DIGRAPHS
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 241 - 243
Copyright
Copyright © Applied Probability Trust 1973 

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References

Berge, C. (1965) Programming, Games and Transportation Networks. John Wiley, New York.Google Scholar
Chartrand, G. (1966) A graph-theoretic approach to a communications problem. SIAM J. Appl. Math. 14, 778–781.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
Feller, D. and Harary, F. (1971) Connectivity in graphs. Recent Trends in Graph Theory. (Proc. conf. in New York, 1970). Lecture Notes in Mathematics 186, 105–115. Springer, Berlin.Google Scholar
Harary, F. (1969) Graph Theory. Addision-Wesley, Reading, Mass.CrossRefGoogle Scholar
Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, New Jersey.Google Scholar
Romanovsky, V. I. (1970) Discrete Markov Chains. Walter Noordhoof, Gronagen, Netherlands.Google Scholar