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


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.


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.


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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

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