A normally regular digraph with parameters (v, k, λ, μ) is a directed graph on v vertices whose adjacency matrix A satisfies the equation AAt = kI + λ(A + At) + μ(J − I − A − At). This means that every vertex has out-degree k, a pair of nonadjacent vertices have μ common out-neighbours, a pair of vertices connected by an edge in one direction have λ common out-neighbours and a pair of vertices ...

We consider the problem (MSSS) of nding a strongly connected spanning subgraph with the minimum number of arcs in a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the hamil-tonian cycle problem. We show that the problem is polynomially solvable for quasi-transitive digraphs. We describe the minimum number of arcs in such a spanning subgraph of a qu...

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call (D)=max16 i6 n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1; V2; : : : ; Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and t...

For a nonnegative n× n matrix A, we find that there is a polynomial f(x) ∈ R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative...

In the paper the definition and main properties of a 2D-digraph, namely a directed graph with two kinds of arcs, are introduced. Under the assumption of strong connectedness, the analysis of its paths and cycles is performed, basing on an integer matrix whose rows represent the compositions of all circuits, and on the corresponding row-module. Natural constraints on the composition of the paths...

We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strongly connected subgraph of a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the hamiltonian cycle problem. We characterize the number of arcs in a minimum spanning strong subgraph for digraphs which are either extended semicomplete or semicomplete bipartite. Our ...

The global dynamics of coupled systems of differential equations defined on an interaction network are investigated. Local dynamics at each vertex, when interactions are absent, are assumed to be simple: solutions to each vertex system are assumed to converge to an equilibrium, either on the boundary or in the interior of the feasible region. The interest is to investigate the collective behavi...

For a strongly connected digraphD the restricted arc-connectivity λ′(D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D − S has a non trivial strong component D1 such that D − V (D1) contains an arc. Let S be a subset of vertices of D. We denote by ω+(S) the set of arcs uv with u ∈ S and v ∈ S, and by ω−(S) the set of arcs uv with u ∈ S and v ∈ S. A dig...

Competition graphs were rst introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices (except K 2) which is the competitiongraph ...