Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Contribution Biplots

To interpret the biplot, it is necessary to know which points—usually variables—are the ones that are important contributors to the solution, especially when there are many variables involved. This information can be calculated separately as part of the biplot's numerical results, but this means that a table has to be consulted along with the graphical display. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic information directly into the display itself, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses, such as correspondence analysis, principal component analysis, log-ratio analysis, and various forms of discriminant analysis, and, in fact, to any method based on dimension reduction through the singular value decomposition. In the contribution biplot, one set of points, usually the rows of a data matrix, optimally represents the spatial positions of the cases or sample units, according to an appropriate distance measure. The other set of points, usually the columns of the data matrix, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that often only one common scale for the row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, the contribution biplot also solves the problem in correspondence analysis and log-ratio analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution. This article has supplementary materials online.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.     

Perturbing convex multiobjecttve programs

The sensitivity of a convex noniinear constrained multiobjective program to small perturbations is analysed, in terms of the stability of weak optimal faces, and vertices, to perturbations. These results are applied to the perturbation of a set-valued optimization problem, in which the objective and constraint set-functions take convex poiyhedra as their values     

Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G=K_k(K₁+K_n−k−1), the join of K_k, the complete graph on k vertices, with the disjoint union of K₁ and K_n−k−1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum.     

Connectivity of the Mycielskian of a digraph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This work investigates the vertex connectivity and arc connectivity of the Mycielskian of a digraph D. This generalizes the recent results due to Balakrishnan and Raj [R. Balakrishnan, S.F. Raj, Connectivity of the Mycielskian of a graph, Discrete Math, 308 (2008), 2607–2610].     

Strongly indexable graphs and applications

In 1990, Acharya and Hegde introduced the concept of strongly k-indexable graphs: A (p,q)-graph G=(V,E) is said to be strongly k-indexable if its vertices can be assigned distinct numbers 0,1,2,…,p−1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices form an arithmetic progression k,k+1,k+2,…,k+(q−1). When k=1, a strongly k-indexable graph is simply called a strongly indexable graph. In this paper, we report some results on strongly k-indexable graphs and give an application of strongly k-indexable graphs to plane geometry, viz; construction of polygons of same internal angles and sides of distinct lengths.