We derive an asymptotic formula for the number of strongly connected digraphs with n vertices and m arcs (directed edges), valid for m − n → ∞ as n → ∞ provided m = O(n log n). This fills the gap between Wright’s results which apply to m = n+O(1), and the long-known threshold for m, above which a random digraph with n vertices and m arcs is likely to be strongly connected.

We attempt to generalize a theorem of Nash-Williams stating that graph has k-arc-connected orientation if and only it is 2k-edge-connected. In strongly connected digraph we call an arc deletable its deletion leaves digraph. Given 3-edge-connected G, define Frank number f(G) be the minimum k such there exist orientations G with property every edge becomes in at least one these orientations. are ...

We propose an O(min{m+n logn,m log∗n}) to find a minmax strongly connected spanning subgraph of a digraph with n nodes and m arcs. A generalization of this problem called the minmax strongly connected subgraph problem with node penalties is also considered. An O(m logn) algorithm is proposed to solve this general problem. We also discuss ways to improve the average complexity of this algorithm.

In this paper we investigate new sufficient conditions for a digraph to be kernel-perfect (KP) and some structural properties of kernel-perfect critical (KPC) digraphs. In particular, it is proved that the asymmetrical part of any KPC digraph is strongly connected. A new method to construct KPC digraphs is developed. The existence of KP and KPC digraphs with arbitrarily large dichromatic number...

Suppose that D = (V,E) is a strongly connected digraph. Let u, v ∈ V (D). The maximum distance md(u, v) is defined as md(u, v)=max{ −→ d (u, v), −→ d (v, u)} where −→ d (u, v) denote the length of a shortest directed u− v path in D. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph D are defined with respect to this metric. The main aim of this paper is ...

A strongly connected digraph D of order n is primitive (aperiodic) provided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t, called the exponent of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t. The exponent of D is the largest of n ‘generalized exponents’ that may be a...

From the theory of Ho0man polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satis3es certain polynomial equation AP(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coe5cients, and Jn is the n×n matrix of all ones. In this paper we present some su5cient conditions, in terms of the coe5cients of P(x), to ensure that ...

Generalizing the idea of viewing a digraph as a model of a linear map, we suggest a multi-variable analogue of a digraph, called a hydra, as a model of a multi-linear map. Walks in digraphs correspond to usual matrix multiplication while walks in hydras correspond to the tensor multiplication introduced by Robert Grone in 1987. By viewing matrix multiplication as a special case of this tensor m...