In this report, we examine the generalization of the Laplacian of a graph due to Fan Chung. We show that Fan Chung’s generalization reduces to examining one particular symmetrization of the adjacency matrix for a directed graph. From this result, the directed Cheeger bounds trivially follow. Additionally, we implement and examine the benefits of directed hierarchical spectral clustering empiric...

Let G denote a directed graph with adjacency matrix Q and indegree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observ...

This work studies the edge-based representation of directed acyclic graphs, as well as the properties of recombination operators working on it. It is shown that this representation is not separable, and the structure of the basic information units that must be processed in order to maintain feasibility of the solutions is described. As an experimental analysis indicates, a recombination operato...

We present a linear time algorithm to greedily orient the edges of a path graph model to obtain a directed path graph model (when possible). Moreover we extend this algorithm to find an odd sun when the method fails. This algorithm has several interesting consequences concerning the relationship between path graphs and directed path graphs. One is that for a directed path graph, path graph mode...

Rank-width is a graph complexity measure that has many structural properties. It is known that the rank-width of an undirected graph is the maximum over all induced prime graphs with respect to split decomposition and an undirected graph has rank-width at most 1 if and only if it is a distance-hereditary graph. We are interested in an extension of these results to directed graphs. We give sever...

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i...

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i...

Treewidth of an undirected graph measures how close the graph is to being a tree. Several problems that are NP-hard on general graphs are solvable in polynomial time on graphs with bounded treewidth. Motivated by the success of treewidth, several directed analogues of treewidth have been introduced to measure the similarity of a directed graph to a directed acyclic graph (DAG). Directed treewid...

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on th...