Spanning Forests of a Digraph and Their Applications

Authors:
R. P. Agaev;P. Yu. Chebotarev
Affiliations:
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia;Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
Venue:
Automation and Remote Control
Year:
2001

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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.