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 ...

An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices is one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r(u) = v, Such that there are two walks of lenght ≤ k from u to v. The smallest positive integer p such that the compositio...

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...

Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such...

We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we constr...

Let G=(V,A) be a digraph. The eccentricity e(u) of a vertex u is the maximum distance from u to any other vertex in G. A vertex v in G is an eccentric vertex of u if the distance from u to v equals e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G and has arcs from a vertex v to its eccentric vertices. In this paper we present several results on the eccentric digraph...

An upward planar drawing of a digraph G is a planar drawing of G where every edge is drawn as a simple curve monotone in the vertical direction. A digraph is upward planar if it has an embedding that admits an upward planar drawing. The problem of testing whether a digraph is upward planar is NP-complete. In this paper we give a linear-time algorithm to test the upward planarity of a series-par...

A minimum reversing set of a digraph is a smallest sized set of arcs which when reversed makes the digraph acyclic. We investigate a related issue: Given an acyclic digraph D, what is the size of a smallest tournataent T which has the arc set of D as a minimum reversing set? We show that such a T always exists and define the reversing number ofan acyclic digraph to be the number of vertices in ...

A Cayley digraph G = C(Γ, X) for a group Γ and a generating set X is the digraph with vertex set V (G) = Γ and arcs (g, gx) where g ∈ Γ and x ∈ X. The reverse of C(Γ, X) is the Cayley digraph G−1 = C(Γ, X−1) where X−1 = {x−1;x ∈ X}. We are interested in sufficient conditions for a Cayley digraph not to be isomorphic to its reverse and focus on Cayley digraphs of metacyclic groups with small gen...

In this paper we introduce two polytopes that respect a digraph in the sense that for every vector in the polytope every component corresponds to a node and is at least equal to the component corresponding to each successor of this node. The sharing polytope is the set of all elements from the unit simplex that respect the digraph. The fuzzy polytope is the set of all elements of the unit cube ...