The principle of inclusion-exclusion is specialized in order to count labeled digraphs with separately speciied out-components, in-components, and isolated components. Applications include counting digraphs with no in-nodes or out-nodes, digraphs with a source and a sink, and digraphs with a unique source and a unique sink.

For a digraph $D$, the $p$-competition graph $C_{p}(D)$ of $D$ is satisfying following: $V(C_{p}(D))=V(D)$, for $x,y \in V(C_{p}(D))$, $xy E(C_{p}(D))$ if and only there exist distinct $p$ vertices $v_{1},$ $v_{2},$ $...,$ $v_{p}$ $\in$ $V(D)$ such that $x \rightarrow v_{i}$, $y v_{i}$ $A(D)$ each $i=1,2,$ $p$.

In this paper, D = (V (D), A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V (D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u → v → w → z in D, then u and z are adjacent. In [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3quasi-transitive digraphs are the ...

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and Hadamard matrices. Commutative non-commutative association schemes shown to be attached the divisible digraphs.

It is known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k ~ 3 there are no digraphs of order 'close' to, i.e., one less than, the Moore bound (14). For diameter 2, it is known that digraphs close to Moore bound exist for any degree because the line digraphs of complete digraphs are an example of such di...

In this paper we introduce a superclass of split digraphs, which we call spine digraphs. Those are the digraphs D whose vertex set can be partitioned into two sets X and Y such that the subdigraph induced by X is traceable and Y is a stable set. We also show that Linial’s Conjecture holds for spine digraphs.

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, `) and it is derived from the...

It is known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see 20] or 5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order`close' to, i.e., one less than, Moore bound 18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of s...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N , u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k− 1)-kernel. This work is a survey of results proving sufficient conditions for the exist...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...