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 prov...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V (D)−N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D, a semikernel modulo F , S of D is an independent set of vertices of ...

Abstract It is known that the spectral radius of a digraph with k edges is ≤ √ k, and that this inequality is strict except when k is a perfect square. For k = m + l, l fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex and connecting it to the first l/2 vertices by pairs of directed edges. Using a combinatori...

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. Expressions are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff matrix. These expressions involve the matrix ...

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkabl...

Given a finite Abelian group G and a generator subset A ⊂ G of cardinality two, we consider the Cayley digraph Γ = Cay(G, A). This digraph is called 2–Cayley digraph. An extension of Γ is a 2–Cayley digraph, Γ′ = Cay(G′, A) with G < G′, such that there is some subgroup H < G′ satisfying the digraph isomorphism Cay(G′/H, A) ∼= Cay(G, A). We also call the digraph Γ a quotient of Γ′. Notice that t...

We study the convergence to stationarity for random walks on dynamic digraphs with given degree sequences. The undergo full regeneration at independent geometrically distributed time intervals parameter α. Relaxation is result of an interplay and mixing static digraph. When number vertices n tends infinity α zero, we find three scenarios according whether αlogn converges or some finite positive...

It is known that regular digraphs of degree d, diameter k and unique walks of length not smaller than h and not greater than k between all pairs of vertices ([ h, k ]-digraphs), exist only for h k and h = k 1, if d ;::: 2. This paper deals with the problem of the enumeration of [k 1, kJ-digraphs in the case of diameter k = 2 or degree d = 2. It is shown, using algebraic techniques, that the lin...