We consider the random directed graph G→(n,p) with vertex set {1,2,…,n} in which each of n(n−1) possible edges is present independently probability p. are interested strongly connected components this graph. A phase transition for emergence a giant component known to occur at p=1/n, critical window p=1/n+λn−4/3 λ∈R. show that, within window, G→(n,p), ranked decreasing order size and rescaled by...

To decide if the parameterized feedback vertex set problem in directed graph is fixed-parameter tractable is a long time open problem. In this paper, we prove that the parameterized feedback vertex set in directed graph is fixed-parameter tractable and give the first FPT algorithm of running time O((1.48k)n) for it. As the feedback arc set problem in directed graph can be transformed to a feedb...

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph C∗-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an AF algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph w...