Let $D$ be a finite and simple digraph with vertex set $V(D)$‎.‎A signed total Roman $k$-dominating function (STR$k$DF) on‎‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

Dominance relations between individuals can be represented by a directed social network. A relational power measure is a function that assigns to each position in a directed network a value representing the relational power of that position in the network. We axiomatically characterize two such power measures, the b-measure and the score-measure. We also apply these measures to weighted directe...

let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

Let D = (V, A) be a directed graph of order p. A subset S of the vertex set V(D) is a dominating set of D if for each vertex v∈D – S there exists a vertex u∈S such that (u, v) is an arc of D. The domination number of D, γ(D), is the order of a smallest dominating set of D. In this paper we calculate the domination number of the cartesian product of two directed paths Pm and Pn for general m and n.

Topological centrality is a significant measure for characterising the relative importance of a node in a complex network. For directed networks that model dynamic processes, however, it is of more practical importance to quantify a vertex's ability to dominate (control or observe) the state of other vertices. In this paper, based on the determination of controllable and observable subspaces un...

The concept of connectedness plays an important role in many networks. Digraphs are considered as an excellent modeling tool and are used to model many types of relations amongst any physical situations. In this paper the concept of strong non-split domination in directed graph D has been introduced by considering the dominating set S is a strong non-split dominating set if the complement of S ...