Event-Triggered Bipartite Consensus Tracking and Vibration Control of Flexible Timoshenko Manipulators Under Time-Varying Actuator Faults

For bipartite angle consensus tracking and vibration suppression of multiple Timoshenko manipulator systems with time-varying actuator faults,parameter and modeling uncertainties,and unknown disturbances,a novel distributed boundary event-triggered control strategy is proposed in this work.In contrast to the earlier findings,time-varying consensus tracking and actuator defects are taken into account simultaneously.In addition,the constructed event-triggered control mechanism can achieve a more flexible design because it is not required to satisfy the input-to-state condition.To achieve the control objectives,some new integral control variables are given by using back-stepping technique and boundary control.Moreover,adaptive neural networks are applied to estimate system uncertainties.With the proposed event-triggered scheme,control inputs can reduce unnecessary updates.Besides,tracking errors and vibration states of the closed-looped network can be exponentially convergent into some small fields,and Zeno behaviors can be excluded.At last,some simulation examples are given to state the effectiveness of the control algorithms.

Bipartite consensus problems of Lurie multi-agent systems over signed graphs: A contraction approach

This paper examines the bipartite consensus problems for the nonlinear multi-agent systems in Lurie dynamics form with cooperative and competitive communication between different agents. Based on the contraction theory, some new conditions for the nonlinear Lurie multi-agent systems reaching bipartite leaderless consensus and bipartite tracking consensus are presented. Compared with the traditional methods, this approach degrades the dimensions of the conditions, eliminates some restrictions of the system matrix, and extends the range of the nonlinear function. Finally, two numerical examples are provided to illustrate the efficiency of our results.

A New Proof on the Bipartite Turán Number of Bipartite Graphs

The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].

Practical bipartite consensus for multi-agent systems:A barrier function-based adaptive sliding-mode control approach

This paper is concerned with bipartite consensus tracking for multi-agent systems with unknown disturbances.A barrier function-based adaptive sliding-mode control(SMC)approach is proposed such that the bipartite steady-state error is converged to a predefined region of zero in finite time.Specifically,based on an error auxiliary taking neighboring antagonistic interactions into account,an SMC law is designed with an adaptive gain.The gain can switch to a positive semi-definite barrier function to ensure that the error auxiliary is constrained to a predefined neighborhood of zero,which in turn guarantees practical bipartite consensus tracking.A distinguished feature of the proposed controller is its independence on the bound of disturbances,while the input chattering phenomenon is alleviated.Finally,a numerical example is provided to verify the effectiveness of the proposed controller.

Cycle Multiplicity of Total Graph of Complete Bipartite Graph

Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bipartite graph, this generalizes the result for, which is given by M.M. Akbar Ali in [1].

Event-Triggered Asymmetric Bipartite Consensus Tracking for Nonlinear Multi-Agent Systems Based on Model-Free Adaptive Control

In this paper,an asymmetric bipartite consensus problem for the nonlinear multi-agent systems with cooperative and antagonistic interactions is studied under the event-triggered mechanism.For the agents described by a structurally balanced signed digraph,the asymmetric bipartite consensus objective is firstly defined,assigning the agents'output to different signs and module values.Considering with the completely unknown dynamics of the agents,a novel event-triggered model-free adaptive bipartite control protocol is designed based on the agents'triggered outputs and an equivalent compact form data model.By utilizing the Lyapunov analysis method,the threshold of the triggering condition is obtained.Subsequently,the asymptotic convergence of the tracking error is deduced and a sufficient condition is obtained based on the contraction mapping principle.Finally,the simulation example further demonstrates the effectiveness of the protocol.

The Evolving Bipartite Network and Semi-Bipartite Network Models with Adjustable Scale and Hybrid Attachment Mechanisms

The bipartite graph structure exists in the connections of many objects in the real world, and the evolving modeling is a good method to describe and understand the generation and evolution within various real complex networks. Previous bipartite models were proposed to mostly explain the principle of attachments, and ignored the diverse growth speed of nodes of sets in different bipartite networks. In this paper, we propose an evolving bipartite network model with adjustable node scale and hybrid attachment mechanisms, which uses different probability parameters to control the scale of two disjoint sets of nodes and the preference strength of hybrid attachment respectively. The results show that the degree distribution of single set in the proposed model follows a shifted power-law distribution when parameter r and s are not equal to 0, or exponential distribution when r or s is equal to 0. Furthermore, we extend the previous model to a semi-bipartite network model, which embeds more user association information into the internal network, so that the model is capable of carrying and revealing more deep information of each user in the network. The simulation results of two models are in good agreement with the empirical data, which verifies that the models have a good performance on real networks from the perspective of degree distribution. We believe these two models are valuable for an explanation of the origin and growth of bipartite systems that truly exist.

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K 7,n when7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.

K_(1,p)~k-FACTORIZATION OF COMPLETE BIPARTITE GRAPHS

In this paper, it is shown that a sufficient condition for the existence of a K 1,p k factorization of K m,n , whenever p is a prime number and k is a positive integer, is (1) m≤p kn,(2) n≤p km,(3)p kn-m≡p km-n ≡0(mod( p 2k -1 )) and (4) (p kn-m)(p km-n) ≡0(mod( p k -1)p k×(p 2k -1)(m+n)) .

ORTHOGONAL (g,f)-FACTORIZAFIONS OF BIPARTITE GRAPHS

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two positive integer-valued functions defined on V(G) such that g(x) ≤ f(x) for every vertex x of V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤ dH(x) 5 f(x) for each x ∈ V(H). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {F1, F2,…… ， Fm } and H be a factorization and a subgraph of G, respectively. If F, 1 ≤ i ≤ m， has exactly one edge in common with H, then it is said that ■ is orthogonal to H. It is proved that every bipartite (mg + m - 1, mf - m + 1 )-graph G has a (g, f)-factorization orthogonal to k vertex disjoint m-subgraphs of G if 2-k ≤ g(x) for all x ∈ V(G). Furthermore, it is showed that the results in this paper are best possible.

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

Two conditions for a bipartite graph to be a k-deleted graph

A k-regular spanning subgraph of graph G is called a k-factor of G. Graph G is called a k-deleted graph if G-e has a k-factor for each edge e. A graph G=(X,Y) with bipartition (X,Y) is called a bipartite graph if every edge of G has one endpoint in X and the other in Y.It is proved that a bipartite graph G=(X,Y) with X=Y is a k-deleted graph if and only if kS≤r 1+2r 2+...+k(r k+...+r Δ)-ε(S) for all SX. Using this result we give a sufficient neighborhood condition for a bipartite to be a k-deleted graph.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.

THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？

Generation of a new bipartite coherent-entangled state and its applications

A new bipartite coherent-entangled state is introduced in the two-mode Fock space, which exhibits the properties of both a coherent state and an entangled state. The set of coherent-entangled states makes up a complete and partly nonorthogonal representation. A simple experimental scheme to produce the coherent-entangled state using an asymmetric beamsplitter is proposed. Some applications of the coherent-entangled state in quantum optics are also oresented.

MINIMUM CONGESTION SPANNING TREES IN BIPARTITE AND RANDOM GRAPHS

The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.

Quantum Correlations in a Family of Bipartite Qubit-Qutrit Separable States

Quantum correlations in a family of bipartite separable qubit-qutrit quantum-classical correlated states are investigated by using two popular measures,i.e.,the original quantum discord(OQD)method by Ollivier and Zurek[Phys.Rev.Lett.88(2001)017901]and the measurement-induced disturbance(MID)method by Luo[Phys.Rev.A 77(2008)022301].It is found that both of them are functions of a parameter partially characterizing the concerned states,however,quantum correlations evaluated via the convenient MID method are somewhat overestimated.

Bipartite Graphs with the First and Second Largest Laplacian Spectral Radius

Let Bn^k be the class of bipartite graphs with n vertices and k cut edges. The extremal graphs with the first and the second largest Laplacian spectral radius among all graphs in Bn^K are presented. The bounds of the Laplacian spectral radius of these extremal graphs are also obtained.

Bipartite Bell Inequality and Maximal Violationt

We present new bell inequalities for arbitrary dimensional bipartite quantum systems. The maximal violation of the inequalities is computed. The Bell inequality is capable of detecting quantum entanglement of both pure and mixed quantum states more effectively.

Bipartite Formation Tracking for Multi-Agent Systems Using Fully Distributed Dynamic Edge-Event-Triggered Protocol

In this study,the bipartite time-varying output formation tracking problem for heterogeneous multi-agent systems(MASs)with multiple leaders and switching commu-nication networks is considered.Note that the switching communication networks may be connected or disconnected.To address this problem,a novel reduced-dimensional observer-based fully distributed asynchronous dynamic edge-event-triggered output feedback control protocol is developed,and the Zeno behavior is ruled out.The theoretical analysis gives the admissible switching frequency and switching width under the proposed control protocol.Different from the existing works,the control protocol reduces the dimension of information to be transmitted between neighboring agents.Moreover,since an additional positive internal dynamic variable is introduced into the triggering functions,the control protocol can guarantee a larger inter-event time interval compared with previous results.Finally,a simulation example is given to verify the effectiveness and performance of the theoretical result.