Diffuse interface methods for multiclass segmentation of high-dimensional data

We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg–Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman–Bence–Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples.     

Distance-hereditary graphs are clique-perfect

In this paper, we show that the clique-transversal number τ_C(G) and the clique-independence number α_C(G) are equal for any distance-hereditary graph G. As a byproduct of proving that τ_C(G)=α_C(G), we give a linear-time algorithm to find a minimum clique-transversal set and a maximum clique-independent set simultaneously for distance-hereditary graphs.     

带警戒限的均值控制图中平均链长的计算公式

带警戒限的均值控制图中 ,平均链长ARL(AverageRunLength)是其重要特性。本文利用转移概率流图TPFG(TransitionProbabilityFlowGraphs)方法导出了关于ARL的一般公式     

Eigenvalues of graphs with threefold symmetry

In many cases, the spectrum of one graph contains the entire spectrum of a second, smaller graph. The larger (composite) graph and the smaller (component) graph are said to be subspectral. Rules are given for constructing two subspectral components of a composite graph which has threefold symmetry such that the eigenvalues of one component and the eigenvalues of the second component taken twice comprise the complete spectrum of the composite graph. The mathematical basis for these rules is shown to be a unitary transformation upon the eigenvalue equation of the adjacency matrix of the composite graph by the matrix which represents threefold rotation.     

On the width-length inequality

This paper was written at the Rensselaer Polytechnic Institute in 1963. A lecture based on it was given at the Rand Corporation in 1965 and this version is in the form in which Ray Fulkerson received it at Rand.The paper is the underpinning for results on resistor network inequalities (Reference [4]) which has not been published. A specific example, however, appears inProceedings of the IEEE 51 (1963) 1047–1048.More is known about minimal non-W—L matrices. U. Peled has found several additional classes of matrices which are also classically known in other contexts. A consequence of a recent matrix theorem is apparently that the degenerate projection planes are the only minimal matrices requiring unequal weights.The preparation of this paper was supported by the National Science Foundation under grant GP-14. Editor's Note: This paper, although written in 1963, was sought for inclusion inMathematical Programming Study 8—Polyhedral Combinatorics, which was dedicated to the memory of D.R. Fulkerson. Unfortunately, an editorial mishap prevented its inclusion. Nevertheless, the historical importance of the paper, the fact that it has been widely referenced and influenced Fulkerson's and others' work in the area has convinced the Editor that it should be published and hence made readily available. Alfred Lehman has given his assent to this despite his reluctance to publish a paper which is not current.     

A divide-and-conquer bound for aggregate’s quality and algebraic connectivity

We establish a divide-and-conquer bound for the aggregate’s quality and algebraic connectivity measures, as defined for weighted undirected graphs. Aggregate’s quality is defined on a set of vertices and, in the context of aggregation-based multigrid methods, it measures how well this set of vertices is represented by a single vertex. On the other hand, algebraic connectivity is defined on a graph, and measures how well this graph is connected. The considered divide-and-conquer bound for aggregate’s quality relates the aggregate’s quality of a union of two disjoint sets of vertices to the aggregate’s quality of the two sets. Likewise, the bound for algebraic connectivity relates the algebraic connectivity of the graph induced by a union of two disjoint sets of vertices to the algebraic connectivity of the graphs induced by the two sets.     

Frugal, acyclic and star colourings of graphs

Given a graph G=(V,E), a vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind et al. (1997) [14] studied proper t-frugal colourings and Yuster (1998) [22] studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study.     

On the Arithmetical Rank of the Edge Ideals of Forests

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.