Let G be a weighted directed graph where the edge weights are non-negative real numbers. An arborescence or (directed) spanning tree with root r is a connected subgraph of G such that there is a unique directed path from v to r for each v ∈ V (G). The weight of a spanning tree T will be the product of the weights of the edges of T . Let Υ(G) be the probability distribution on the set of all spa...

The topological ordering algorithm sorts nodes of a directed graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed graph. Then, we apply the topological orderingalgorithm on graphs containing the covering nodes. We show that there exists a cut set withforward arcs in these...

Navarro, Frenk, & White have recently found numerically that the density profile of dark-matter halos can be described by a universal two-parameter function over a broad range of halo masses. The profile is singular, approaching the halo center with ρ ∝ r−1. It had been argued previously that radially distorted, gravitationally lensed images of background sources in galaxy clusters, so-called r...