Probability Distribution of Edge in Adjacent Matrix of Aviation Network of China and Algorithm of Searching Non-overlap Community Structure Based on Complex Network

In order to discover the probability distribution feature of edge in aviation network adjacent matrix of China and on the basis of this feature to establish an algorithm of searching non-overlap community structure in network to reveal the inner principle of complex network with the feature of small world in aspect of adjacent matrix and community structure,aviation network adjacent matrix of China was transformed according to the node rank and the matrix was arranged on the basis of ascending node rank with the center point as original point.Adjacent probability from the original point to extension around in approximate area was calculated.Through fitting probability distribution curve,power function of probability distribution of edge in adjacent matrix arranged by ascending node rank was found.According to the feature of adjacent probability distribution,deleting step by step with node rank ascending algorithm was set up to search non-overlap community structure in network and the flow chart of algorithm was given.A non-overlap community structure with 10 different scale communities in aviation network of China was found by the computer program written on the basis of this algorithm.

两类四全图的奇异性

Let G be a finite simple graph and A(G)be its adjacency matrix.Then G is singular if A(G)is singular.The graph obtained by bonding the starting ver-tices and ending vertices of three paths Pa1,Pa2,Pa3 is calledθ-graph,represented byθ(a1,a2,a3).The graph obtained by bonding the two end vertices of the path Ps to the vertices of theθ(a1,a2,a3)andθ(b1,b2,b3)of degree three,respectively,is denoted byα(a1,a2,a3,s,b1,b2,b3)and calledα-graph.β-graph is denoted whenβ(a1,a2,a3,b1,b2,b3)=α(a1,a2,a3,1,b1,b2,b3).In this paper,we give the necessary and sufficient conditions for the singularity ofα-graph andβ-graph,and prove that the probability that a random givenα-graph andβ-graph is a singular graph is equal to 14232048 and 733/1024,respectively.

The Angles and Main Angles of Some Special Graphs

The eigenvalues of the adjacency matrix of a graph are called the eigenvalues of the graph. Let the vector ej =(0, … , 1, … , 0)T and the all -1 vector j =(1, 1, …,1)T, the cosine of the (acute) angle formed by the vector ej and the eigensubspace is called an angle of the graph. The cosine of the (acute) angle formed by the vector j and the eigensubspace is called a main angle of the graph. The angles and main angles are all important parameters on the graph, and they can be combined with the eigenvalues of the graph to determine the degree sequence of the graph, the number of triangles, quadrilaterals and pentagons on the graph, and the characteristic polynomials of the complement graph, but there is little study on the angles and main angles of the graph. In this paper, we determine the angles and main angles of the complete graph, the cube graph, the Petersen graph, the cycle and the complete bipartite graph.

Computer generation of detailed reaction networks in hydrocracking of Fischer-Tropsch wax

Fischer-Tropsch synthesis(FTS)wax is a mixture of linear hydrocarbons with carbon number from C7 to C70+.Converting FTS wax into high-quality diesel(no sulfur and nitrogen contents)by hydrocracking technology is attractive in economy and practicability.Kinetic study of the hydrocracking of FTS wax in elementary step level is very challenging because of the huge amounts of reactions and species involved.Generation of reaction networks for hydrocracking of FTS wax in which the chain length goes up to C70 is described on the basis of Boolean adjacency matrixes.Each of the species(including paraffins,olefins and carbenium ions)involved in the elementary steps is represented digitally by using a(N+3)N matrix,in which a group of standardized numbering rules are designed to guarantee the unique identity of the species.Subsequently,the elementary steps are expressed by computer-aided matrix transformations in terms of proposed reaction rules.Dynamic memory allocation is used in species storage and a characteristic vector with nine elements is designed to store the key information of a(N+3)N matrix,which obviously reduces computer memory consumption and improves computing efficiency.The detailed reaction networks of FTS wax hydrocracking can be generated smoothly and accurately by the current method.The work is the basis of advanced elementary-step-level kinetic modeling.

Topological Configuration and Rotation Analysis of the Three⁃Order Rubik􀆳s Cube

The mechanism of three⁃order Rubik􀆳s Cube(RC)has the characteristics of recombination and variable degree of freedom,and it is difficult to accurately describe the degree of its freedom.This paper takes RC as the research object,and the adjacency matrix is constructed based on topology and graph theory in order to describe the variation rule of topological configuration in the single layer rotation of RC.In this paper,the degree of freedom of the RC in any shape can be described by defining the concept of entanglement degree of freedom,establishing a set of adjacency matrix,and determining the degree of freedom of the RC which is attributed to the number of non⁃zero elements in the set of adjacent matrix.The prime number is proposed to describe the rotation of the RC combined with the rotation recognition of RC,which is simple and convenient for computer processing.The research contents in this paper are beneficial to the study of RC from the perspective of mechanism science.Meanwhile,it is of great significance to the study of other complex mechanisms with variable degrees of freedom.

The Spectral Radii of Some Adhesive Graphs

The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.

Review and Simulation of a Novel Formation Building Algorithm While Enabling Obstacle Avoidance

Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to eliminate the risk of human injuries while working on hazardous tasks, and to improve productivity. Globally, engineers are continuously researching better, simple, and faster cooperative Control algorithms to provide a Control strategy where each agent in the robot formation can communicate effectively and achieve a consensus in their position, orientation and speed. This paper explores a novel Formation Building Algorithm and its global stability around a configuration vector. A simulation in MATLAB? was carried out to examine the performance of the Algorithm for two geometric formations and a fixed number of robots. In addition, an obstacle avoidance technique was presented assuming that all robots are equipped with range sensors. In particular, a uniform rounded obstacle is used to analyze the performance of the technique with the use of detailed geometric calculations.

On the Number of Cycles in a Graph

In this paper, we obtain explicit formulae for the number of 7-cycles and the total number of cycles of lengths 6 and 7 which contain a specific vertex vi in a simple graph G, in terms of the adjacency matrix and with the help of combinatorics.

Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties

The generating function for generating integer sequence of Aunu numbers of prime cardinality was reported earlier by the author in [1]. This paper assigns an operator on the function for where the operation induces addition or subtraction on the pairs of ai, aj elements which are consecutive pairs of elements obtained from a generating set of some finite order. The paper identifies that the set of the generated pairs of integer sequence is non-associative. The paper also presents the graph theoretic applications of the integers generated in which subgraphs are deduced from the main graph and adjacency matrices and incidence matrices constructed. It was also established that some of the subgraphs were found to be regular graphs. The findings in this paper can further be used in coding theory, Boolean algebra and circuit designs.

A Topological Clustering of Variables

The clustering of objects(individuals or variables)is one of the most used approaches to exploring multivariate data.The two most common unsupervised clustering strategies are hierarchical ascending clustering(HAC)and k-means partitioning used to identify groups of similar objects in a dataset to divide it into homogeneous groups.The proposed topological clustering of variables,called TCV,studies an homogeneous set of variables defined on the same set of individuals,based on the notion of neighborhood graphs,some of these variables are more-or-less correlated or linked according to the type quantitative or qualitative of the variables.This topological data analysis approach can then be useful for dimension reduction and variable selection.It’s a topological hierarchical clustering analysis of a set of variables which can be quantitative,qualitative or a mixture of both.It arranges variables into homogeneous groups according to their correlations or associations studied in a topological context of principal component analysis(PCA)or multiple correspondence analysis(MCA).The proposed TCV is adapted to the type of data considered,its principle is presented and illustrated using simple real datasets with quantitative,qualitative and mixed variables.The results of these illustrative examples are compared to those of other variables clustering approaches.

Space truss construction modeling based on on-orbit assembly motion feature

More space truss construction has been planned to develop and utilize space resources.These trusses are designed in the way of large-scale,complex,modular,and on-orbit assembly.To meet the upcoming challenge of large-scale space infrastructure construction,it is necessary to study space truss automation design and robotic construction.This paper proposes an ordinal finite screw adjacency matrix model(OFSAMM),focusing on the relationship between assembly motions,to express and compute a space truss structure.In this model,a space truss is abstracted as a set of ordered assembly motions,each of which is recorded as a finite screw as the basic element of the truss and its assembly.The operation of truss transformation is also derived under this model.Therefore,the truss configuration,the assembly sequence,the truss sub-assembly,the truss components,and the on-orbit assembly task can be expressed and calculated in a unified model,which is calculated and stores the truss topology and assembly with the minimum storage cost.At the end of this paper,we introduce how to synthesize and optimize space truss design through two cases.The study will help to improve design efficiency.Furthermore,it provides a theoretical basis for the automatic construction of space truss structures,especially in the next stage.

Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namely,for any Cayley digraph X(D2P,T),X(D2P,T)and X(D2P,S)having the same H-spectrum implies that they are isomorphic.

Revealing connectivity in residential Architecture: An algorithmic approach to extracting adjacency matrices from floor plans

In today's world, various approaches and parameters exist for designing a plan and determining its spatial, placement. Hence, various modes for identifying crucial locations can be explored when an architectural plan is designed in different dimensions. While designing all these modes takes considerable time, there are numerous potential applications for artificial intelligence (AI) in this domain. This study aims to compute and use an adjacency matrix to generate architectural residential plans. Additionally, it develops a plan generation algorithm in Rhinoceros software, utilizing the Grasshopper plugin to create a dataset of architectural plans. In the following step, the data was entered into a neural network to identify the architectural plan's type, furniture, icons, and use of spaces, which was achieved using YOLOv4, EfficientDet, YOLOv5, DetectoRS, and RetinaNet. The algorithm's execution, testing, and training were conducted using Darknet and PyTorch. The research dataset comprises 12,000 plans, with 70% employed in the training phase and 30% in the testing phase. The network was appropriately trained practically and precisely in relation to an average precision (AP) resulting of 91.50%. After detecting the types of space use, the main research algorithm has been designed and coded, which includes determining the adjacency matrix of architectural plan spaces in seven stages. All research processes were conducted in Python, including dataset preparation, network object detection, and adjacency matrix algorithm design. Finally, the adjacency matrix is given to the input of the proposed plan generator network, which consequently, based on the resulting adjacency, obtains different placement modes for spaces and furniture.

On the Least Eigenvalue of Graphs with Cut Vertices

Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.

The Least Eigenvalue of Unicyclic Graphs with Application to Spectral Spread

Let U^(g)_(n)be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),…,T_(g))∈U^(g)_(n)be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in an anticlockwise direction)by identifying v_(i)with the root of a rooted tree T_(i)of order n_(i)for each i=1,2,…,g,where n_(i)≥1 and∑^(g)_(i=1)n_(i)=n.In this note,the graph with the minimal least eigenvalue(and the graph with maximal spread)in C(T_(1),T_(2),…,T_(g))is determined.

Minimal least eigenvalue of connected graphs of order n and size m=n+k(5≤k≤8)

The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix.We characterize the connected graphs of order n and size n+k(5≤k≤8 and n≥k+5)with the minimal least eigenvalue.

A new fractional order 6D chaotic model:Study of model dynamics,system structure graph,electronic circuit realization and fractional control

A new fractional 6D chaotic model is constructed in this paper.The new fractional 6D chaotic model has six positive parameters plus the fractional order with eight nonlinear terms.The complicated chaotic dy-namics of the new fractional 6D model is presented and analyzed.The basic properties of this model are studied and its chaotic attractors,dissipative feature,symmetry,equilibrium points,Lyapunov Exponents are investigated.The new dynamics of the 6D fractional model is numerically simulated using Matlab software.In addition,utilizing the graph theory tools certain structural characteristics are calculated.An electrical circuit is built to implement the new 5.4 fractional order 6D model.Finally,an active fractional order controller is proposed to control the new model at different fractional orders.The chaos of the new model is very useful and can be used to produce random keys for data encryption.