Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Hernandez, J.M., van Mieghem, P.: Classification of graph metrics, p. 20. Delft https://www.nas.ewi.tudelft.nl/people/Piet/papers/TUDreport20111111_MetricList.pdf (2011). Accessed 27 March 2019
Gantmakher, F.R.: Matrix Theory, 3rd edn, p. 576. Nauka, Moscow (1967)
Kemeny, J., Snell, J.: Finite Markov Chains. University Series in Undergraduate Mathematics, p. 210. Van Nostrand, Princeton (1960)
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)
Hu, T.C.: Integer Programming and Network Flows, p. 452. Addison-Wesley, Menlo Park-London (1970)
Ore, O.: Theory of Graphs, p. 270. American Mathematical Society, Providence US (1962)
Davies, D., Barber, D.: Communication Networks for Computers, p. 575. Wiley, Hoboken (1973)
Diestel, R.: Graph Theory Graduate Texts in Mathematics, vol. 173, p. 322. Springer-Verlag, New York (2000)
Van Dooren, P.: Graph Theory and Applications, p. 110. Dublin. https://perso.uclouvain.be/paul.vandooren/DublinCourse.pdf (2009). Accessed 27 March 2019
Ruohonen, K.: Graph Theory, p. 114. Tampere University of Technology. http://math.tut.fi/~ruohonen/GT_English.pdf (2013). Accessed 27 March 2019
Zykov, A.A.: Fundamentals of Graph Theory, p. 382. Nauka, Moscow (1987)
Wilson, R.J.: Introduction to Graph Theory, 4th edn, p. 171. Edinburgh Gate, Harlow and Essex, Addison Wesley Longman Limited (1996)
Harary, F.: Graph Theory, p. 274. Addison-Wesley, Reading (1969)
Anderson, J.A.: Discrete Mathematics with Combinatorics, 1st edn, p. 799. Prentice Hall, Upper Saddle River (2000)
Biggs, N.: Algebraic Graph Theory, 2nd edn, p. 211. Cambridge University Press, Cambridge-New York-Melbourne (1993)
Godsil, C., Royle, G.: Algebraic graph theory. In: Graduate Texts in Mathematics, vol. 207, p. 443. Springer-Verlag, New York (2001)
Idjiry, Y.: On the generalized inverse of an incidence matrix. J. SIAM 13(3), 827–836 (1965)
Boullion, T.L., Odell, P.L.: Generalized Inverse Matrices, p. 107. Wiley Interscience, New York-London-Sydney-Toronto (1971)
Albert, A.E.: Regression and the Moore-Penrose Pseudoinverse, p. 180. Academic Press, New York (1972)
Beklemishev, D.V.: Additional Chapters of Linear Algebra, p. 336. Nauka, Moscow (1983)
Rusakov, V.A.: Reconstruction of the Euclidian Metric of an Undirected Graph by Metrics of Components. Nat. Tech. Sci. 2(52), 22–24 (2011)
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)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Rusakov, V.A. (2020). On Markov Chains and Some Matrices and Metrics for Undirected Graphs. In: Antipova, T. (eds) Integrated Science in Digital Age. ICIS 2019. Lecture Notes in Networks and Systems, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-030-22493-6_30
Download citation
DOI: https://doi.org/10.1007/978-3-030-22493-6_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22492-9
Online ISBN: 978-3-030-22493-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)