Types Of Vertices Research Articles

Geometrical defects in two-dimensional melting of many-particle Yukawa systems

We present a theoretical polygon construction analysis of two-dimensional melting and freezing transitions in many-particle Yukawa systems. Two-dimensional melting transitions can be characterized as proliferation of geometrical defects--nontriangular polygons, obtained by removing unusually long bonds in the triangulation of particle positions. A liquid state is characterized by the temperature-independent number of quadrilaterals and linearly increasing number of pentagons. We analyze specific types of vertices, classified by the type and distribution of polygons surrounding them, and determine temperature dependencies of their concentrations. Solid-liquid phase transitions are followed by the peaks in the abundances of certain types of vertices.

Bond topology and structure-generating functions: graph-theoretic prediction of chemical composition and structure in polysomatic T–O–T (biopyribole) and H–O–H structures

AbstractAspects of the bond topology and chemical composition of a mineral may be incorporated into a general formula by writing the local topological details of each cation and anion, along with their chemical identity, as a general expression called a structure-generating function. Here, this procedure is described for polysomatic T–O–T and H–O–H structures. We may write tetrahedrally coordinated cations and their associated anions as {T2nΘm}. For {T2nΘm} to be a chain or ribbon, 5n < m ≤ 6n, and we may write m as 5n + N, where N is an integer. Within the {T2nΘ(5n+N)}unit, we may recognize three types of anion vertices: (1) bridging anions, Θbr, that are bonded to two T cations; (2) apical anions, Θap, that are involved in linkage to other cations out of the plane of the bridging anions; and (3) linking anions, Θl, that link to non-T cations in the plane of the bridging anions. We may incorporate the connectivity of the cations in our algebraic representation of the chain as follows: {T2nΘbraΘapbΘlc} where a + b + c = 5n + N. The apical anions of the T- or H-sheets provide some anions of the layer of octahedra. We may use the handshaking di-lemma of graph theory to examine the interaction between the two types of layers, and write aStructure-Generating Function, S(N;n), that gives both the stoichiometry and aspects of the bond topology of the structures.Where N = 1, the T-sheet consists of ribbons of the form {T2nΘ(5n+1)} = {T2nΘbr(3n–1)Θap2nΘl2}. Each T–Θbr–T linkage spans an octahedron, and hence there are (3n – 1) octahedrally coordinated cations between opposing {T2nΘbr(3n–1)Θap2nΘl2} ribbons. There are an additional (n–1) vertices, Ψ, required to complete the coordination of the M cations on one side of the O-sheet, and we may write the structure-generating function for biopyriboles as follows: S(1;n)= Xi[M(3n–1)Ψ2(n–1){T2nΘbr(3n–1)Θap2nΘl2}2] = [M(3n–1)Ψ2(n–1){T2nΘ(5n+1)}2]. Where N = 2, the general form of the T-ribbon is {T2nΘ(5n+2)}, a component of the H-sheet in the polysomatic H–O–H minerals in which the T-ribbons are linked laterally by [5]- or [6]-coordinated high-valence cations, D, which have the coordination (Dφ41φapφt), where ftmay or may not be present depending on the coordination number, [6] or [5], of the D cation. The general formula for an H-sheet is [Dφap{T2nΘbr(3n–2)Θap2nΘl4}φt0–1], where φt(written after the T-sheet) occurs on the outside of the H-sheet and may be involved in linkage between adjacent H–O–H blocks. The H-sheet links via its apical anions to the O-sheet, giving the general formula of an H–O–H block as [M(3n+1)(DφapΨn{T2nΘ(5n+2)}φt0–1)2]. These H–O–H blocks may link directly or indirectly through the φt anions of the (DΘl4φapφt) octahedra, giving S(2;n)= Xi[M(3n+1)Ψ2n(D2φap2{T2nΘbr(3n–2)Θap2nΘl4}2)φt0–2]. Combining the expressions for the structure-generating functions gives a single function for T–O–T and H–O–H structures:S(N;n)= Xi[M(3n+2N–3)?2(n+N–2)(D2(N–1)f2ap(N–1){T2nT(3n–N)brT2napT2N1}2)f0–2(N–1)t]This expression also generates mixed-ribbon polysomatic structures. Thus S(1;2+3)gives the chemical composition and structure of the mixed-chain pyribole chesterite, and S(2;1+4)gives the chemical composition and structure of the mixed-chain H–O–H mineral, veblenite.

A novel gray image representation using overlapping rectangular NAM and extended shading approach

In this paper, inspired by the idea of overlapping rectangular region coding of binary images, we extend the SDS design, which is based on overlapping representation from binary images to gray images based on the non-symmetry and anti-packing model (NAM). A novel gray image representation is proposed by using the overlapping rectangular NAM (RNAM) and the extended Gouraud shading approach, which is called ORNAM representation. Also, we present an ORNAM representation algorithm of gray images. The encoding and the decoding of the proposed algorithm can be performed in O(n logn) time and O(n) time, respectively, where n denotes the number of pixels in a gray image. The wrong decoding problem of the hybrid matrix R for the overlapping RNAM representation of gray images is solved by using the horizontal, vertical, and isolated matrices, i.e., H, V and I, respectively, which are used to identify the vertex types. Also, we put forward four criteria of anti-packing homogeneous blocks. In addition, by redefining a codeword set for the three vertices symbols, we also propose a new coordinate data compression procedure for coding the coordinates of all non-zone elements in the three matrices H, V and I. By taking some idiomatic standard gray images in the field of image processing as typical test objects, and by comparing our proposed ORNAM representation with the conventional S-Tree Coding (STC) representation, the experimental results in this paper show that the former has higher compression ratio and less number of homogeneous blocks than the latter whereas maintaining a satisfactory image quality, and therefore it is a better method to represent gray images.

Types of maximum noninteger vertices of the relaxation polyhedron of the four-index axial assignment problem

We describe various types of maximum noninteger vertices. We identify types of polyhedron vertices by the number of fractional components contained in three-sections of fourindex matrices representing the polyhedron vertices.

On the Minimum ABC Index of Chemical Trees

The Atom Bond Connectivity index, also known as ABC index was defined by Estrada(4) with relation to the energy of formation of alkanes. It was quickly recognized that this index reflects important structural properties of graphs in general. The ABC index was extensively studied in the last three years, from the point of view of chemical graph theory(5,6), and in general graphs(1). It was also compared to other structural indices of graphs(2). Das derives multiple results with implications to the minimum/maximum ABC index on graphs. With relation to trees, it is known that among all the trees of the same number of vertices, the maximum ABC index is attained for the star graph. However, it is not known which tree(s) minimize(s) the ABC index. The problem seems to be hard. It is partially addressed in many sources(5,1,6), but remains open. In this paper we further investigate the trees that minimize the ABC index. Our investigations are limited to chemical trees, i.e. trees in which the maximum vertex degree is 4. The chemical trees were introduced to reflect the structure of the carbon chains and the molecules based on them. Our approach is algorithmic. We identify certain types of edges (chemical bonds) that are important and occur frequently in chemical trees. Further, we study how the removal of a certain edge, the intro- duction of certain edge or the contraction of certain edge affects the ABC-index of the tree. We pay particular attention to the examples of minimal ABC index chemical trees provided by Dimitrov(3). This paper will discuss the Atom-bond connectivity index, its origins, its uses and its applications. The central topics of discussion for this paper will include: an investigation of the Furtula examples (provided through a personal communica- tion), an exploration of the structure of the maximal and minimal chemical trees for the ABC index, and the contri- bution of certain types of vertices and edges to the ABC index of the graph as a whole. It will also consider graphs from a specific class Γ (to be defined later) and their ABC indices.

Connectivity editing for quadrilateral meshes

We propose new connectivity editing operations for quadrilateral meshes with the unique ability to explicitly control the location, orientation, type, and number of the irregular vertices (valence not equal to four) in the mesh while preserving sharp edges. We provide theoretical analysis on what editing operations are possible and impossible and introduce three fundamental operations to move and re-orient a pair of irregular vertices. We argue that our editing operations are fundamental, because they only change the quad mesh in the smallest possible region and involve the fewest irregular vertices (i.e., two). The irregular vertex movement operations are supplemented by operations for the splitting, merging, canceling, and aligning of irregular vertices. We explain how the proposed high-level operations are realized through graph-level editing operations such as quad collapses, edge flips, and edge splits. The utility of these mesh editing operations are demonstrated by improving the connectivity of quad meshes generated from state-of-art quadrangulation techniques.

Polygon construction to investigate melting in two-dimensional strongly coupled dusty plasma

The polygon construction method of Glaser and Clark is used to characterize melting and crystallization in a two-dimensional (2D) strongly coupled dusty plasma. Using particle positions measured by video microscopy, bonds are identified by triangulation, and unusually long bonds are deleted. The resulting polygons have three or more sides. Geometrical defects, which are polygons with more than three sides, are found to proliferate during melting. Pentagons are found in liquids, where they tend to cluster with other pentagons. Quadrilaterals are a less severe defect, so that disorder can be characterized by the ratio of quadrilaterals to pentagons. This ratio is found to be less in a liquid than in a solid or a superheated solid. Another measure of disorder is the abundance of different kinds of vertices, according to the type of polygons that adjoin there. Unexpectedly, spikes are observed in the abundance of certain vertex types during rapid temperature changes. Hysteresis, revealed by a plot of a disorder parameter vs temperature, is examined to study sudden heating. The hysteresis diagram also reveals features suggesting a possibility of latent heat in the melting and rapid cooling processes.

Searches on star graphs and equivalent oracle problems

We examine a search on a graph among a number of different kinds of objects (vertices), one of which we want to find. In a standard graph search, all of the vertices are the same, except for one, the marked vertex, and that is the one we wish to find. We examine the case in which the unmarked vertices can be of different types, so the background against which the search is done is not uniform. We find that the search can still be successful, but the probability of success is lower than in the uniform background case, and that probability decreases with the number of types of unmarked vertices. We also show how the graph searches can be rephrased as equivalent oracle problems.

Eight different decagonal tilings derived from rhombic Penrose tiling

We found 32 variations of dot tiling (DT) reported previously [J. Alloy. Comp. 342 (2002) p.206], which is similar to a tiling consisting of pentagon, hexagons, nonagon and decagons (PHND) [Phil. Mag. A l76 (1997) p.85] employed in the structure analysis of decagonal quasicrystals. They are its superstructures, with a five times larger unit cell in four-dimensional space. These tilings are built up by adding one dot in each fat rhombus in the rhombic Penrose tiling. Eight tilings among 32 DTs do not break the local symmetries of the eight types of vertices in the rhombic Penrose tiling. The eight tilings and their occupation domains are shown.

Randomised Reproducing Graphs

We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, $\alpha$, $\beta$ and $\gamma$, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If $(1+\alpha)(1+\gamma)<1$ then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on $\alpha$ and $\gamma$. If $(1+\alpha)(1+\gamma)>1$ then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree $d$ tends to zero. We also give some results on the number of edges and on the spectral gap.

Editing operations for irregular vertices in triangle meshes

We describe an interactive editing framework that provides control over the type, location, and number of irregular vertices in a triangle mesh. We first provide a theoretical analysis to identify the simplest possible operations for editing irregular vertices and then introduce a hierarchy of editing operations to control the type, location, and number of irregular vertices. We demonstrate the power of our editing framework with an example application in pattern design on surfaces.

Network ensemble clustering using latent roles

We present a clustering method for collections of graphs based on the assumptions that graphs in the same cluster have a similar role structure and that the respective roles can be founded on implicit vertex types. Given a network ensemble (a collection of attributed graphs with some substantive commonality), we start by partitioning the set of all vertices based on attribute similarity. Projection of each graph onto the resulting vertex types yields feature vectors of equal dimensionality, irrespective of the original graph sizes. These feature vectors are then subjected to standard clustering methods. This approach is motivated by social network concepts, and we demonstrate its utility on an ensemble of personal networks of migrants, where we extract structurally similar groups and show their resemblance to predicted acculturation strategies.

A NEW MODULARITY FOR DETECTING ONE-TO-MANY CORRESPONDENCE OF COMMUNITIES IN BIPARTITE NETWORKS

Real-world relations are often represented as bipartite networks, such as paper-author networks and event-attendee networks. Extracting dense subnetworks (communities) from bipartite networks and evaluating their qualities are practically important research topics. As the attempts for evaluating divisions of bipartite networks, Guimera and Barber propose bipartite modularities. This paper discusses the properties of these bipartite modularities and proposes another bipartite modularity that allows one-to-many correspondence of communities of different vertex types.

Degree Distributions in General Random Intersection Graphs

We study $G(n,m,F,H)$, a variant of the standard random intersection graph model in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights, the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type.

Primordial non-Gaussianities from the trispectra in multiple field inflationary models

We investigate the primordial non-Gaussianities from the trispectra in multi-field inflation models, which can be seen as generalization of multi-field k-inflation and multi-DBI inflation. We derive the full fourth-order perturbation action for the inflaton fields and evaluate the four-point correlation functions for the perturbations in the limit ca << 1 and ce << 1. There are three types of momentum-dependentshape functions which arise from three types of four-pointinteraction vertices. The final trispectrum of the curvature perturbation can be expressed in terms of the deformations and permutations of these three shape functions, and is determined by ca, ce, λ , Π which depend on the non-linear structure of the model and also the transfer function Tℛ\U0001d4ae. We also discuss the parameter space for the trispectrum and plot the shape diagrams for the trispectrum both for visualization and for distinguishing different shapes from each other.

Restricted square-triangle tilings

Abstract Restricted square-triangle tilings are tilings of regular squares and triangles where the number of vertex types has been limited to a subset of all possible vertex types. Such structures have recently been observed in several places: for example as strip phases in colloidal patterns on laser fields with pentagonal symmetry, and as square-pair-free tilings in patterns formed by dipolar magnetic binary colloidal mixtures. In general there are four vertex configurations: A (4 squares, 44), Z (6 triangles, 36), H (4233), and S (43433). Strip phases belong to type AHZ, square-pair-free phases to SZ, standard inflation generated quasicrystals to HSZ, and unrestricted dodecagonal random tilings to AHSZ. We study several vertex type combinations and describe the properties of the patterns generated. AHZ tilings can be mapped to binary necklaces and classified completely. A similar mapping is possible for many HS tilings. Defect-free SZ tilings can be transformed into a triangle-square-hexagon (TSH) supertiling with matching rules. Monte-Carlo (MC) simulations are used to examine the properties of several classes of restricted tilings.

Multi-scaling Networks from Vertex Intrinsic Fitness

Defining and detecting the community structure is an important topic on exploring the complex networks. Previous works were mostly based on the so-called modularity method, in which vertices that interconnect to each other form modules in the network. In many real-world networks, however, vertices are grouped not by their connectivity but by their functions. To demonstrate this idea, we propose a new kind of network, in which the vertices are catalogued into several types and are assigned intrinsic fitness. Each type of vertices may satisfy a different fitness distribution. It is found that the whole network exhibits a multi-scaling degree distribution. The clustering coefficients of the subnetworks are modified by the interlinks between vertices in the subnetworks.

Scatter networks: a new approach for analysing information scatter

Information on any given topic is often scattered across the Web. Previously this scatter has been characterized through the inequality of distribution of facts (i.e. pieces of information) across webpages. Such an approach conceals how specific facts (e.g. rare facts) occur in specific types of pages (e.g. fact-rich pages). To reveal such regularities, we construct bipartite networks, consisting of two types of vertices: the facts contained in webpages and the webpages themselves. Such a representation enables the application of a series of network analysis techniques, revealing structural features such as connectivity, robustness and clustering. Not only does network analysis yield new insights into information scatter, but we also illustrate the benefit of applying new and existing analysis techniques directly to a bipartite network as opposed to its one-mode projection. We discuss the implications of each network feature to the users’ ability to find comprehensive information online. Finally, we compare the bipartite graph structure of webpages and facts with the hyperlink structure between the webpages.

Using a genetic algorithm to study properties of minimum energy states and geometrical frustration in artificial “spin ice” systems

This article reports the results of a study on the base state of artificially frustrated “spin ice” systems. We have studied the states of minimum energy reported by experimental studies on nanoscale ferromagnetic islands and the protocols employed to reach those states. The main technique employed in this study is a genetic algorithm that has been contrasted with two Montecarlo methods. Nanoscale islands are modeled through dipolar moments placed on a plane, rectangular array. Studies include the correlation between nanoscale islands, statistics on vertex types formed in the array for the minimum energy state and intermediate states. The results suggest a failure in the protocols adopted to minimize energy in these systems. A study on the efficiency between the devised genetic algorithm and the Montecarlo methods used in the research is also included.

On a phase diagram for random neural networks with embedded spike timing dependent plasticity

This paper presents an original mathematical framework based on graph theory which is a first attempt to investigate the dynamics of a model of neural networks with embedded spike timing dependent plasticity. The neurons correspond to integrate-and-fire units located at the vertices of a finite subset of 2D lattice. There are two types of vertices, corresponding to the inhibitory and the excitatory neurons. The edges are directed and labelled by the discrete values of the synaptic strength. We assume that there is an initial firing pattern corresponding to a subset of units that generate a spike. The number of activated externally vertices is a small fraction of the entire network. The model presented here describes how such pattern propagates throughout the network as a random walk on graph. Several results are compared with computational simulations and new data are presented for identifying critical parameters of the model.