On Markov Chains and Some Matrices and Metrics for Undirected Graphs

  • Victor A. RusakovEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 78)


Metric tasks often arise as a simplification of complex and practically important problems on graphs. The correspondence between the search algorithms of the usual shortest paths and Markov chains is shown. From this starting point a sequence of matrix descriptions of undirected graphs is established. The sequence ends with the description of the explicit form of the Moore-Penrose pseudo inversed incidence matrix. Such a matrix is a powerful analytical and computational tool for working with edge flows with conditionally minimal Euclidian norms. The metrics of a graph are represented as its characteristics generated by the norms of linear spaces of edge and vertex flows. The Euclidian metric demonstrates the advantages of the practice of solving problems on graphs in comparison with traditional metrics based on the shortest paths or minimal cuts.


Markov chains The Moore-Penrose pseudo inverse of the incidence matrix Shortest paths Minimal cuts The Euclidian metric on graphs 


  1. 1.
    Kozyrev, V.P., Yushmanov, S.V.: Graph theory (algorithmic, algebraic and metric problems). The results of science and technology. Probability theory. Math. Stat. Theor. Cybern. Ser. 23, 68–117 (1985)zbMATHGoogle Scholar
  2. 2.
    Hernandez, J.M., van Mieghem, P.: Classification of graph metrics, p. 20. Delft (2011). Accessed 27 March 2019
  3. 3.
    Gantmakher, F.R.: Matrix Theory, 3rd edn, p. 576. Nauka, Moscow (1967)Google Scholar
  4. 4.
    Kemeny, J., Snell, J.: Finite Markov Chains. University Series in Undergraduate Mathematics, p. 210. Van Nostrand, Princeton (1960)Google Scholar
  5. 5.
    Rusakov, V.A.: Analysis and Synthesis of Computer Network Structures. Part 1. Analysis, p 122. Moscow Engg. Phys. Inst. Report: All-Union Sci. Tech. Inform. Center No. Б796153 (1979)Google Scholar
  6. 6.
    Hu, T.C.: Integer Programming and Network Flows, p. 452. Addison-Wesley, Menlo Park-London (1970)Google Scholar
  7. 7.
    Ore, O.: Theory of Graphs, p. 270. American Mathematical Society, Providence US (1962)Google Scholar
  8. 8.
    Davies, D., Barber, D.: Communication Networks for Computers, p. 575. Wiley, Hoboken (1973)Google Scholar
  9. 9.
    Diestel, R.: Graph Theory Graduate Texts in Mathematics, vol. 173, p. 322. Springer-Verlag, New York (2000)Google Scholar
  10. 10.
    Van Dooren, P.: Graph Theory and Applications, p. 110. Dublin. (2009). Accessed 27 March 2019
  11. 11.
    Ruohonen, K.: Graph Theory, p. 114. Tampere University of Technology. (2013). Accessed 27 March 2019
  12. 12.
    Zykov, A.A.: Fundamentals of Graph Theory, p. 382. Nauka, Moscow (1987)Google Scholar
  13. 13.
    Wilson, R.J.: Introduction to Graph Theory, 4th edn, p. 171. Edinburgh Gate, Harlow and Essex, Addison Wesley Longman Limited (1996)zbMATHGoogle Scholar
  14. 14.
    Harary, F.: Graph Theory, p. 274. Addison-Wesley, Reading (1969)CrossRefGoogle Scholar
  15. 15.
    Anderson, J.A.: Discrete Mathematics with Combinatorics, 1st edn, p. 799. Prentice Hall, Upper Saddle River (2000)Google Scholar
  16. 16.
    Biggs, N.: Algebraic Graph Theory, 2nd edn, p. 211. Cambridge University Press, Cambridge-New York-Melbourne (1993)Google Scholar
  17. 17.
    Godsil, C., Royle, G.: Algebraic graph theory. In: Graduate Texts in Mathematics, vol. 207, p. 443. Springer-Verlag, New York (2001)Google Scholar
  18. 18.
    Idjiry, Y.: On the generalized inverse of an incidence matrix. J. SIAM 13(3), 827–836 (1965)MathSciNetGoogle Scholar
  19. 19.
    Boullion, T.L., Odell, P.L.: Generalized Inverse Matrices, p. 107. Wiley Interscience, New York-London-Sydney-Toronto (1971)zbMATHGoogle Scholar
  20. 20.
    Albert, A.E.: Regression and the Moore-Penrose Pseudoinverse, p. 180. Academic Press, New York (1972)zbMATHGoogle Scholar
  21. 21.
    Beklemishev, D.V.: Additional Chapters of Linear Algebra, p. 336. Nauka, Moscow (1983)Google Scholar
  22. 22.
    Rusakov, V.A.: Reconstruction of the Euclidian Metric of an Undirected Graph by Metrics of Components. Nat. Tech. Sci. 2(52), 22–24 (2011)Google Scholar
  23. 23.
    Rusakov, V.A.: A Technique for Analyzing and Synthesizing the Structures of Computer Networks Using Markov Chains. Computer Networks and Data Transmission Systems, pp. 62–68. Znaniye, Moscow (1977)Google Scholar
  24. 24.
    Rusakov, V.A.: Implementation of the methodology for analysis and synthesis of computer network structures using Markov chains. Engineering-mathematical methods in physics and cybernetics. Issue 7, pp. 41–45. Atomizdat, Moscow (1978)Google Scholar
  25. 25.
    Rusakov, V.A.: Synthesis of computer network structures and the problem of small certainty of initial values. USSR AS’s Scientific Council on Cybernetics. In: Proceedings of 5th All-Union School-Seminar on Computing Networks, 1, pp. 112–116. VINITI, Moscow-Vladivostok (1980)Google Scholar
  26. 26.
    Rusakov, V.A.: On the regularity of the displacement of the mean estimate for the throughput with non-stationary traffic. USSR AS’s Scientific Council on Cybernetics. In: Proceedings of 9th All-Union School-Seminar on Computing Networks, 1.2, pp. 48–52. VINITI, Moscow-Pushchino (1984)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia

Personalised recommendations