This paper proposes a simple but effective graph-based agglomerative algorithm, for clustering high-dimensional data. We explore the different roles of two fundamental concepts in graph theory, indegree and outdegree, in the context of clustering. The average indegree reflects the density near a sample, and the average outdegree characterizes the local geometry around a sample. Based on such in...

Let N = {1, . . . , n} be a finite set. Let E be a collection of ordered or unordered pairs of distinct 1 elements from N . A graph G is defined by (N,E). The elements of N are called vertices of graph G. The elements of E are called edges of graph G. If E consists of ordered pairs of vertices, then the graph is called a directed graph (digraph). When we say graph, we refer to undirected graph,...

Woodall’s conjecture asserts the following about every directed graph: if every directed cut of the graph has k or more edges then the graph has k or more mutually disjoint dijoins. Here, a dijoin is a set J of arcs such that any vertex is connected to any other by a path all of whose forward-directed arcs are in J . This talk is a little survey of the counterexamples to a generalized version o...

In 1960 Ghouila-Houri extended Dirac’s theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed grap...

This work presents an algorithm for computing the maximum flow and minimum cut of undirected graphs, based on the well-known algorithm presented by Ford and Fulkerson for directed graphs. The new algorithm is equivalent to just applying Ford and Fulkerson algorithm to the directed graph obtained from original graph but with two directed arcs for each edge in the graph, one in each way. We prese...

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that almost all the regular Gaussian distributions that factorize with respect to a chain graph are faithful to it. This result has three important consequences. First, chain graphs are more powerful than undirected graphs and acyclic directed graphs for representing regular Gaussian distr...

There is a long-standing debate about the principles constraining the combinatorial properties of affixes. Hay (2002) and Hay and Plag (2004) proposed a model in which suffixes can be ordered along a hierarchy of processing complexity. We show that their model generalizes to a larger set of suffixes, and we provide independent evidence supporting their claim that a higher rank in the ordering c...

Recently, kernel methods have been widely employed to solve machine learning problems such as classification and clustering. Although there are many existing graph kernel methods for comparing patterns represented by undirected graphs, the corresponding methods for directed structures are less developed. In this paper, to fill this gap in the literature we exploit the graph kernels and graph co...

Consider the random graph process where we start with an empty graph on n vertices, and at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that whp the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph has a directe...

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most dn(g−2)+1 g−1 e. When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albert...