In this paper the properties of node-node adjacency matrix in acyclic digraphs are considered. It is shown that topological ordering and node-node adjacency matrix are closely related. In fact, first the one to one correspondence between upper triangularity of node-node adjacency matrix and existence of directed cycles in digraphs is proved and then with this correspondence other properties of ...

The orientable domination number, DOM(G), of a graph G is the largest number over all orientations G. In this paper, DOM studied on different product graphs and related operations. arbitrary corona products determined, while sharp lower upper bounds are proved for Cartesian lexicographic products. A result Chartrand et al. (1996) extended by establishing values DOM(Kn1,n2,n3) positive integers ...

 It is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. In this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.  

We study the problem of enumerating all rooted directed spanning trees (arborescences) of a directed graph (digraph) G = (V,E) of n vertices. An arborescence A consisting of edges e1, . . . , en−1 can be represented as a monomial e1 · e2 · · · en−1 in variables e ∈ E. All arborescences arb(G) of a digraph then define the Kirchhoff polynomial ∑ A∈arb(G) ∏ e∈A e. We show how to compute a compact ...

Eternal domination is a problem that asks the following question: Can we eternally defend graph ? The principle to against an attacked vertex, changes every turn, by moving guards along edges of graph. In classical version game (Burger et al., 2004), only one guard can move at time, but in m-eternal (Goddard 2005), any number single turn. This led introduction two parameters: eternal and number...

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

Assuming that states will remain a permanent feature of our world, what is the ideal that we should hold out for the international order? An attractive proposal is that those peoples that are already organized under non-dominating, representative states should pursue a twin goal: first, arrange things so that they each enjoy the republican ideal of freedom as non-domination in relation to one a...

let $d$ be a digraph with skew-adjacency matrix $s(d)$. then the skew energyof $d$ is defined to be the sum of the norms of all eigenvalues of $s(d)$. denote by$mathcal{o}_n$ the class of digraphs on order $n$ with no even cycles, and by$mathcal{o}_{n,m}$ the class of digraphs in $mathcal{o}_n$ with $m$ arcs.in this paper, we first give the minimal skew energy digraphs in$mathcal{o}_n$ and $mat...