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 directed triple system of order v, denoted DTS(v), is said to be f-bicyclic if it admits an automorphism consisting of f fixed points and two disjoint cycles. In this paper, we give necessary and sufficient conditions for the existence of f-bicyclic

Bridgehead bicyclic unsaturated enamines were prepared by a tandem rhodium-catalyzed C-H bondactivation/alkenylation/electrocyclization of alkyne-tethered unsaturated imines. These strained bicyclic enamines exhibitunique reactivity: for example, they give N-Alkylated products upon treatment with alkylating reagents and undergo doublebond isomerization to alleviate ring strain upon ...

Among unicyclic digraphs and signed digraphs with fixed number of vertices, the digraphs and signed digraphs with minimal and maximal energy are already determined. In this paper we address the problem of finding bicyclic signed digraphs with extremal energy. We obtain minimal and maximal energy among all those n -vertex bicyclic signed digraphs which contain vertex-disjoint signed directed cyc...

Given a molecular graph G, the Hosoya index Z(G) of G is defined as the total number of the matchings of the graph. Let Bn denote the set of bicyclic graphs on n vertices. In this paper, the minimal, the second-, the third-, the fourth-, and the fifth-minimal Hosoya indices of bicyclic graphs in the set Bn are characterized.

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...