Computing Topological Invariants of Triangular Chandelier Lattice

A numerical parameter mathematically derived from the graph structure is a topological index. The topological index is the first actual choice in QSAR research and these indices are used to build the correlation model between the chemical structures of various chemicals compounds. Here, we investigate some old degree-based topological indices like Randic index, sum connectivity index, ABC index, GA index, 1^st and 2^nd Zagreb indices, modified second Zagreb index, redefined version of 1^st, 2^nd and 3^rd Zagreb indices, hyper and augmented Zagreb indices, forgotten index and symmetric division degree index, and some new degree-based indices like SK index, SK₁ index, SK₂ index, and AG₁ index of triangular chandelier-lattice (TCL). The results are generalized by using edge partition and closed formulas for topological indices of triangular chandelier-lattice are analysed.     

Compact Mathematical Formulation for Graph Partitioning

The graph partitioning problem consists of dividing the vertices of a graph into clusters, such that the weight of the edges crossing between clusters is minimized. We present a new compact mathematical formulation of this problem, based on the use of binary representation for the index of clusters assigned to vertices. This new formulation is almost minimal in terms of the number of variables and constraints and of the density of the constraint matrix. Its linear relaxation brings a very fast computational resolution, compared with the standard one.Experiments were conducted on classical large benchmark graphs designed for comparing heuristic methods. On one hand, these experiments show that the new formulation is surprisingly less time efficient than expected on general k-partitioning problems. On the other hand, the new formulation applied on bisection problems allows to obtain the optimum solution for about ten instances, where only best upper bounds were previously known.     

完全图的剖分图的线图的谱

如果一个图G的邻接矩阵A(G)的特征多项式的所有特征值全为整数,则称图G是整的.设图L2(Kp):L(s(Kp))是完全图Kp的剖分图S(Kp)的线图.在这篇文章里,我们利用图的理论给出了S(Kp)和L2(Kp)的特征多项式及其谱.对于图L2(Kp),得到了其补图、线图、线图的补图及补图的线图的特征多项式.也证明了这些图都是整图.这些整图的发现是对整图的研究的一个新贡献.     

Graph theoretic algorithm for automatic operation sequencing for progressive die design

The paper presents a formal graph representation scheme for stamping operation sequencing for sheetmetal progressive dies and a graph theoretic method for automatic determination of the stamping operation sequence. Operation relations are represented by two graphs: an operation precedence graph and an adjacency graph. The first represents precedence constraints defined by best manufacturing practices. The second represents geometric infeasibility relations between operations. Each operation is a member of both graphs. The graphs are automatically generated from a set of stamping operations, which in turn are associated with stamping features of a part. The operation precedence graph is then verified to be acyclic using a coloured Depth First Search. Based on the operation precedence graph, a modified topological sort algorithm is applied to cluster the operations into partially ordered sets. Finally, a graph-colouring algorithm is applied to the operation adjacency graph on the partially ordered operation sets. The algorithm is implemented in C++ and is fully integrated with SolidWorks computer-aided design system. A case study is presented to illustrate the algorithm.     

条形图和圆环图数据占比映射的比较研究

目的 研究在观察时间受限制的条件下,条形图和圆环图在映射数据占总体比值关系时的优势。方法 实验设计了3种尺寸（占屏幕7.5%,15%,22%）和3种映射取值（1,1/2,1/3）的条形图和圆环图,被测对象对图形映射的数据占比进行判断,以获得2种形态的判断时间均值和错误率均值。实验使用计算机化实验生成系统E-prime收集数据,使用SPSS进行数据统计和分析。结果 映射取值为1、1/2、1/3条件下,条形图的3种尺寸判断时间均值均低于圆环图;条形图判断错误率均值为2.51%,低于圆环图的错误率4.03%;2种图的映射取值越小,判断的速度越慢。结论 条形图的映射判断速度和准确性比圆环图更具有优势,更适合应用于时空受限制的场景,有助于数据可视化设计时减少因数据堆集而导致的信息过载。     

Node-pair reliability of network systems with small distances between adjacent nodes

A new method for computing the node-pair reliability of network systems modeled by random graphs with nodes arranged in sequence is presented. It is based on a recursive algorithm using the “sliding window” technique, the window being composed of several consecutive nodes. In a single step, the connectivity probabilities for all nodes included in the window are found. Subsequently, the window is moved one node forward. This process is repeated until, in the last step, the window reaches the terminal node. The connectivity probabilities found at that point are used to compute the node-pair reliability of the network system considered. The algorithm is designed especially for graphs with small distances between adjacent nodes, where the distance between two nodes is defined as the absolute value of the difference between the nodes’ numbers. The maximal distance between any two adjacent nodes is denoted by Γ(G), where G symbolizes a random graph. If Γ(G)=2 then the method can be applied for directed as well as undirected graphs whose nodes and edges are subject to failure. This is important in view of the fact that many algorithms computing network reliability are designed for graphs with failure-prone edges and reliable nodes. If Γ(G)=3 then the method's applicability is limited to undirected graphs with reliable nodes. The main asset of the presented algorithms is their low numerical complexity—O(n), where n denotes the number of nodes.     

Analysis of Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

Chemical compounds are modeled as graphs. The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges. The topological indices representing the molecular graph corresponds to the different chemical properties of compounds. Let be are two positive integers, and be the zero-divisor graph of the commutative ring . In this article some direct questions have been answered that can be utilized latterly in different applications. This study starts with simple computations, leading to a quite complex ring theoretic problems to prove certain properties. The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics, communication theory, cryptography, combinatorics, algorithms analysis, and engineering. In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring (for as prime numbers) with the help of graphical structure analysis. The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graph .     

ABC β–a heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint

This paper deals with the dynamic multi-item capacitated lot-sizing problem (CLSP) with random demand over a finite discrete time horizon. Unfilled demands are backordered. It is assumed that a fill rate constraint is in effect. We propose a heuristic solution procedure called ABC _β that extends the A/B/C heuristic introduced by Maes and Van Wassenhove for the deterministic CLSP to the case of random demands.     

具有最小距离拉普拉斯谱半径的双圈图（英）

一个连通图的距离拉普拉斯矩阵的最大特征值称为这个图的距离拉普拉斯谱半径．本文中,我们先得到距离拉普拉斯谱半径的一个好的下界,然后利用这个下界确定了单圈图中具有最小距离拉普拉斯谱半径的唯一极图．最后,再次利用这个下界,并结合距离拉普拉斯矩阵的特征多项式确定出了双圈图中具有最小距离拉普拉斯谱半径的极图．     

Rhombellation: a builder for multi-shell clusters

Rhombellanes are mathematical structures, proposed by us in 2017; they may appear both in periodic crystals or in finite structures. The simplest rhombellane is rbl.5 or K_2.3, the complete bipartite graph. In this paper, rhombellation operation is iteratively applied on three classes of structures: cube, tori and cubic pcu-network, respectively. The structural topological parameters are detailed.     

基于遗传算法的图关联着色算法

图的着色算法是一种典型的NP-完全问题。给出了一种用于图的关联着色的遗传算法。遗传算法用于进行全局搜索,从而有效的查找解空间。文中对关联色数为6的一个图进行了仿真实验,给出了该图的关联色数以及4种6-关联着色。用本文提出的算法,得到了完全图、完全多部图的关联色数。实验结果表明,本文设计的遗传算法可以很好的对关联着色猜想进行求解,获得问题的高质量的解。     

A simple method for the determination of refractive indices of (rough) transparent solids

Simple methods for the determination of refractive indices of transparent polymers and inorganic and organic solids of irregular geometry or with scratched or corrugated surfaces are rare. A classical procedure is based on the invisibility of a body immersed in a liquid with the same refractive index as that of the body. In order to avoid the laborious procedure connected with the search for a liquid with matching refractive index and to find an approach which is independent of the observation by eye, we describe here a modified immersion method which allows the ready determination of the refractive index of solids. The present method is based on the interpolation of the maximum transmission (n_Tmax) of a solid immersed in liquids with different, typically non-matching, refractive indices. Illustrations with quartz glass, crown glass and poly(vinylidene fluoride) (PVDF) films showed that n_Tmax can be determined with a reproducibility of ± 0.003. By comparison with refractive indices determined by ellipsometry, it was concluded that the refractive index of a solid can be determined with the modified immersion method within an accuracy better than ± 0.01 when systematic errors resulting from the fit method are also taken into consideration.     

细分图的Wiener多项式

本文研究做为最古老的基于距离的拓朴指标之一-Wiener指标.对于Tadpole图Tn,k,循环图Cn,轮式图Wn+1和Helm图Hn+1,文中分别导出了它们的细分图的Wiener多项式的表达式.     

Privacy-Preserving Quantum Two-Party Geometric Intersection

Privacy-preserving computational geometry is the research area on the intersection of the domains of secure multi-party computation (SMC) and computational geometry. As an important field, the privacy-preserving geometric intersection (PGI) problem is when each of the multiple parties has a private geometric graph and seeks to determine whether their graphs intersect or not without revealing their private information. In this study, through representing Alice’s (Bob’s) private geometric graph G_A (G_B) as the set of numbered grids S_A (S_B), an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol is proposed. In the protocol, the oracle operation O_A (O_B) is firstly utilized to encode the private elements of S_A =(a₀,a₁,…,a_M-1) (S_B =(b₀,b₁,…,b_N-1)) into the quantum states, and then the oracle operation O_f is applied to obtain a new quantum state which includes the XOR results between each element of S_A and S_B. Finally, the quantum counting is introduced to get the amount (t) of the states |a_i⊕b_j| equaling to |0|, and the intersection result can be obtained by judging t >0 or not. Compared with classical PGI protocols, our proposed protocol not only has higher security, but also holds lower communication complexity.     

基于图非负矩阵分解的图像配准

本文提出一种新的利用图的谱对应绝对值特征向量的非负矩阵分解图像配准方法.首先利用图像特征构造了无向权图的非负权矩阵,通过非负矩阵分解得到了包含原始图像全部特征的特征基图像;然后将非负权矩阵谱对应绝对值特征向量作为非负矩阵分解的初始值进行迭代,既能反映图的结构特征信息,又能提高图像的匹配率;最后在特征基向量空间找到了两图...     

Graph theory and stability analysis of protein complex interaction networks

Protein complexes play an essential role in many biological processes. Complexes can interact with other complexes to form protein complex interaction network (PCIN) that involves in important cellular processes. There are relatively few studies on examining the interaction topology among protein complexes; and little is known about the stability of PCIN under perturbations. We employed graph theoretical approach to reveal hidden properties and features of four species PCINs. Two main issues are addressed, (i) the global and local network topological properties, and (ii) the stability of the networks under 12 types of perturbations. According to the topological parameter classification, we identified some critical protein complexes and validated that the topological analysis approach could provide meaningful biological interpretations of the protein complex systems. Through the Kolmogorov–Smimov test, we showed that local topological parameters are good indicators to characterise the structure of PCINs. We further demonstrated the effectiveness of the current approach by performing the scalability and data normalization tests. To measure the robustness of PCINs, we proposed to consider eight topological‐based perturbations, which are specifically applicable in scenarios of targeted, sustained attacks. We found that the degree‐based, betweenness‐based and brokering‐coefficient‐based perturbations have the largest effect on network stability.Inspec keywords: graph theory, perturbation theory, proteins, molecular configurations, molecular biophysics, pattern clustering, pattern classification, cellular biophysics, biology computingOther keywords: stability analysis, protein complex interaction networks, biological processes, protein complex interaction network, cellular processes, interaction topology, graph theoretical approach, local network topological properties, topological parameter classification, Kolmogorov‐Smirnov test, network structures, data normalisation tests, topological‐based perturbations, highly clustered protein complexes, brokering‐coefficient‐based perturbations, betweenness‐based perturbations     

Highly effective phototransducers based on polycrystalline heterostructures of AIIBVI compounds

Thin-film polycrystalline heterostructures based on A^IIB^VI compounds were prepared, and related highly effective surface-barrier phototransducers of the p-Cu_1.8S/n-A^IIB^VI/n-A^IIB^VI type (with a quantum efficiency of ∼0.9) were obtained. The heterostructures are promising materials for the creation of solar cells and UV photodetectors insensitive to the visible light.     

Average Co magnetic moment of R-Co-B (R = Y,Pr and Nd) compounds

We estimate the average Co magnetic moment and account for the moment variation by a model in Y_{n + 1}Co_{3n + 5}B_2n (n = 0, 1, 2, 3 and ∞), Pr_{m + n}Co_{5m + 3n}B_2n (m = 2, n = 1 and m = 2, n = 3), Nd_{m + n}Co_{5m + 3n}B_2n (m = 3, n = 2) and Y₂Co₁₄B compounds. And we obtain the magnetic properties of hypothetical Y_{m + n}Co_{5m + 3n}B_2n (n = 1, m = 2, 3, 4 and 5) compounds.