The Matrix of Maximum Out Forests of a Digraph and Its Applications

We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph Γ is a spanning subgraph of Γ that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains out arborescences, then maximum out forests coincide with them. We consider Markov chains related to a weighted digraph and prove that the matrix of Cesàro limiting probabilities of such a chain coincides with the normalized matrix of maximum out forests. This provides an interpretation for the matrix of Cesàro limiting probabilities of an arbitrary stationary finite Markov chain in terms of the weight of maximum out forests. We discuss the applications of the matrix of maximum out forests and its transposition, the matrix of limiting accessibilities of a digraph, to the problems of preference aggregation, measuring the vertex proximity, and uncovering the structure of a digraph.