Advertisement

Automation and Remote Control

, Volume 62, Issue 3, pp 443–466 | Cite as

Spanning Forests of a Digraph and Their Applications

  • R. P. Agaev
  • P. Yu. Chebotarev
Article

Abstract

We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expression are given for the Moore–Penrose generalized inverse and the group inverse of the Kirchhoff matrix. These expressions involve the matrix of maximum out forest of the digraph. Every matrix of out forests with a fixed number of arcs and the normalized matrix of out forests are represented as polynomials of the Kirchhoff matrix; with the help of these identities new proofs are given for the matrix-forest theorem and some other statements. A connection is specified between the forest dimension of a digraph and the degree of an annihilating polynomial for the Kirchhoff matrix. Some accessibility measures for digraph vertices are considered. These are based on the enumeration of spanning forests.

Keywords

Mechanical Engineer Markov Chain System Theory Transition Matrix Fixed Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Harary, F., Norman, R.Z., and Cartwright, D., Structural Models: An Introduction to the Theory of Directed Graphs, New York: Wiley, 1965.Google Scholar
  2. 2.
    Fiedler, M. and Sedláček, J., O W-basích Orientovaných Grafů, Časopis Pěst. Mat., 1958, vol. 83, pp. 214–225.Google Scholar
  3. 3.
    Kelmans, A.K. and Chelnokov, V.M., A Certain Polynomial of a Graph and Graphs with an Extremal Number of Trees, J. Comb. Theory, 1974, ser. B, vol. 16, pp. 197–214.Google Scholar
  4. 4.
    Liu, C.J. and Chow, Yu., Enumeration of Forests in a Graph, Proc. Am. Math. Soc., 1981, vol. 83, pp. 659–663.Google Scholar
  5. 5.
    Chaiken, S., A Combinatorial Proof of the All Minors Matrix Tree Theorem, SIAM J. Discr. Math., 1982, vol. 3, pp. 319–329.Google Scholar
  6. 6.
    Myrvold, W., Counting k-Component Forests of a Graph, Networks, 1992, vol. 22, pp. 647–652.Google Scholar
  7. 7.
    Bapat, R.B. and Constantine, G., An Enumerating Function for Spanning Forests with Color Restrictions, Linear Algebra Appl., 1992, vol. 173, pp. 231–237.Google Scholar
  8. 8.
    Takacs, L., Enumeration of Rooted Trees and Forests, Math. Sci., 1993, vol. 18, pp. 1–10.Google Scholar
  9. 9.
    Erdős, P.L., A New Bijection on Rooted Forests, Discrete Math., 1993, vol. 111, pp. 179–188.Google Scholar
  10. 10.
    Merris, R., Doubly Stochastic Graph Matrices, Univ. Beograd Publ. Elektrotehn. Fak. (Ser. Mat.), 1997, vol. 8, pp. 64–71.Google Scholar
  11. 11.
    Merris, R., Doubly Stochastic Graph Matrices II, Lin. Multilin. Algebra, 1998, vol. 45, pp. 275–285.Google Scholar
  12. 12.
    Chebotarev P.Yu. and Agaev R.P. The Matrix of Maximum Out Forests and Structural Properties of Systems Modeled by Digraphs Modelling and Simulation of Systems MOSIS-2000 34th Spring Int. Conf. Ostrava 2000, vol. 1, pp. 101–106Google Scholar
  13. 13.
    Agaev, R.P. and Chebotarev, P.Yu., The Matrix of Maximum Out Forests and Its Applications, Avtom. Telemekh., 2000, no. 9, pp. 15–43.Google Scholar
  14. 14.
    Chebotarev, P.Yu. and Shamis, E.V., The Matrix-Forest Theorem and Measuring Relations in Small Social Groups, Avtom. Telemekh., 1997, no. 9, pp. 124–136.Google Scholar
  15. 15.
    Chebotarev, P.Yu. and Shamis, E.V., On Proximity Measures for Graph Vertices, Avtom. Telemekh., 1998, no. 10, pp. 113–133.Google Scholar
  16. 16.
    Harary, F., Graph Theory, Reading: Addison-Wesley, 1969.Google Scholar
  17. 17.
    Tutte, W.T., Graph Theory, Reading: Addison-Wesley, 1984.Google Scholar
  18. 18.
    Zykov, A.A., Teoriya konechnykh grafov (Theory of Finite Graphs), Novosibirsk: Nauka, 1969.Google Scholar
  19. 19.
    Faddeev, D.K. and Faddeeva, V.N., Computational Methods of Linear Algebra, San Francisco: Freeman, 1963.Google Scholar
  20. 20.
    Berman, A. and Plemmons, R., Nonnegative Matrices in the Mathematical Sciences, New York: Academic, 1979.Google Scholar
  21. 21.
    Gelfand, I.M., Lektsii po lineinoi algebre (Lectures on Linear Algebra), Moscow: Nauka, 1971.Google Scholar
  22. 22.
    Campbell, S.L. and Meyer, C.D., Generalized Inverses of Linear Transformations, London: Pitman, 1979.Google Scholar
  23. 23.
    Chebotarev, P.Yu. and Shamis, E.V., Preference Fusion when the Number of Alternatives Exceeds Two: Indirect Scoring Procedures, J. Franklin Inst., 1999, vol. 36, pp. 205–226.Google Scholar
  24. 24.
    Hall, K., An r-dimensional Quadratic Placement Problem, Manag. Sci., 1970, vol. 17, pp. 219–229.Google Scholar
  25. 25.
    Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1986.Google Scholar
  26. 26.
    Chebotarev, P.Yu. and Shamis, E.V., On a Duality Between Metrics and Σ-proximities, Avtom. Telemekh., 1998, no. 4, pp. 204–209.Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2001

Authors and Affiliations

  • R. P. Agaev
    • 1
  • P. Yu. Chebotarev
    • 1
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

Personalised recommendations