It is known that regular digraphs of degree d, diameter k and unique walks of length not smaller than h and not greater than k between all pairs of vertices ([ h, k ]-digraphs), exist only for h k and h = k 1, if d ;::: 2. This paper deals with the problem of the enumeration of [k 1, kJ-digraphs in the case of diameter k = 2 or degree d = 2. It is shown, using algebraic techniques, that the lin...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in...

Let $R$ be a commutative ring. In this paper, by using algebraic properties of $R$, we study the Hase digraph of prime ideals of $R$.

Given a digraph D = (V, A), a quasi-kernel of D is an independent set Q C_ V such that for every vertex v not contained in Q, either there exists a vertex u E Q such that v dominates u, or there exists a vertex w such that v dominates w and w dominates some vertex u E Q. A sink in a digraph D = (V, A) is a vertex v E V that dominates no vertex of D. In this thesis we prove that if D is a semico...

A digraph D is called a kernel-perfect digraph or KP -digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V (ζ(D)) = V (D), and A (ζ(D)) ...

In this paper, the concept of continuous multifunctions for digraphs is introduced. It turns out that a digraph multifunction is continuous if and only if it is strong. We introduce the notion of dismantlable digraphs and show that it does generalize the dismantlable graphs and posets. Generalizing various previous results, we show that any dismantlable digraph has the almost fixed point proper...

Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution a PDE, Vlasov Equation (VE). Yet, many applications demand on networks/graphs. In this paper, we interested in sequence directed graphs, or digraphs for short. It interesting to know, how associated with influences macrosco...