A digraph with maximum out-degree d and radius k has at most 1+d+ · · ·+d vertices, as the Moore bound states. Regular digraphs attaining such a bound and whose diameter is at most k + 1 are called radially Moore digraphs. Knor [4] proved that these extremal digraphs do exist for any value of d ≥ 1 and k ≥ 1. In this paper, we introduce a digraph operator, based on the line digraph, which allow...

A weighted composite digraph is obtained from some weighted digraph by replacing each vertex with a weighted digraph. In this paper, we give a beautiful combinatorial proof of the formula for forest volumes of composite digraphs obtained by Kelmans-Pak-Postnikov [7]. Moreover, a generalization of this formula is present.

Given a finite abelian group A, a subset ⊆ A and an endomorphism of A, the endo-Cayley digraph GA( , ) is defined by taking A as the vertex set and making every vertex x adjacent to the vertices (x)+ a with a ∈ . When A is cyclic and the set is of the form = {e, e + h, . . . , e + (d − 1)h}, the digraph G is called a consecutive digraph. In this paper we study the hamiltonicity of endo-Cayley d...

The restricted arc-connectivity λ′ of a strongly connected digraph G is the minimum cardinality of an arc cut F in G such that every strongly connected component of G−F contains at least two vertices. This paper shows that for a d-regular strongly connected digraph with order n and diameter k ≥ 4, if λ′ exists, then λ′(G) ≥ min { (n − dk−1)(d− 1) dk−1 + d− 2 , 2d− 2 } As consequences, the restr...

A digraph obtained by replacing each edge of a complete n-partite (n 2:: 2) graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete n-partite digraph or semicomplete multipartite digraph (abbreviated to SMD). In this paper we show the following result for a semicomplete multipartite digraph of order p with the partite sets VI, 112, ... , Vn. Let r...

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = ∑ h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1, . . . . The sequence {gh(D)|h ≥ 0}, which ...

A characterization of the adjacency matrix of the line digraph of a regular digraph is given. Some corollaries are observed.

Let D be a strongly k−connected digraph of order n ≥ 2. We prove that for every l ≥ n 2k the power Dl of D is Hamiltonian. Moreover, for any n > 2k ≥ 2 we exhibit strongly k-connected digraphs D of order n such that Dl−1 is non-Hamiltonian for l = d n 2ke. We use standard terminology, unless otherwise stated. A digraph D = (V, A) of order n ≥ 2 is said to be strongly q-arc Hamiltonian if for an...

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f(k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f(5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root...