scholarly journals A Matrix Approach to Hypergraph Stable Set and Coloring Problems with Its Application to Storing Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Min Meng ◽  
Jun-e Feng
Keyword(s):  
Vertex Coloring ◽  
Stable Set ◽  
Stable Sets ◽  
Vertex Coloring Problem ◽  
Minimum Coloring ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
The Given ◽  
Product Of Matrices ◽  
Theoretical Results

This paper considers the stable set and coloring problems of hypergraphs and presents several new results and algorithms using the semitensor product of matrices. By the definitions of an incidence matrix of a hypergraph and characteristic logical vector of a vertex subset, an equivalent algebraic condition is established for hypergraph stable sets, as well as a new algorithm, which can be used to search all the stable sets of any hypergraph. Then, the vertex coloring problem is investigated, and a necessary and sufficient condition in the form of algebraic inequalities is derived. Furthermore, with an algorithm, all the coloring schemes and minimum coloring partitions with the givenqcolors can be calculated for any hypergraph. Finally, one illustrative example and its application to storing problem are provided to show the effectiveness and applicability of the theoretical results.

scholarly journals Matrix Approach to Formulate and Search k-ESS of Graphs Using the STP Theory

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Jumei Yue ◽  
Yongyi Yan ◽  
He Deng
Keyword(s):  
Sufficient Conditions ◽  
Complete Information ◽  
Stable Set ◽  
Matrix Product ◽  
Matrix Approach ◽  
Matrix Operation ◽  
Network Systems ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
Product Of Matrices

In this paper, the structure of graphs in terms of k-externally stable set (k-ESS) is investigated by a matrix method based on a new matrix product, called semitensor product of matrices. By defining an eigenvector and an eigenvalue of the node subset of a graph, three necessary and sufficient conditions of k-ESS, minimum k-ESS, and k-kernels of graphs are proposed in a matrix form, respectively. Using these conditions, the concepts of k-ESS matrix, minimum k-ESS matrix, and k-kernel matrix are introduced. These matrices provide complete information of the corresponding structures of a graph. Further, three algorithms are designed, respectively, to find all these three structures of a graph by conducting a series of matrix operation. Finally, the correctness and effectiveness of the results are checked by studying an example. The proposed method and results may offer a new way to investigate the problems related to graph structures in the field of network systems.

scholarly journals Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Yalu Li ◽  
Wenhui Dou ◽  
Haitao Li ◽  
Xin Liu
Keyword(s):  
Matrix Method ◽  
Finite Automata ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Controllability Matrix ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
Product Of Matrices ◽  
Inputs And Outputs

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Finally, an illustrative example is given to support the obtained new results.

scholarly journals Greedoids on vertex sets of B-joins of graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 335-344
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Vertex Sets

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).

scholarly journals On Robust Synchronization of Drive-Response Boolean Control Networks with Disturbances

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yuanyuan Li ◽  
Jie Zhong ◽  
Jianquan Lu ◽  
Zhen Wang ◽  
Fuad E. Alssadi
Keyword(s):  
State Feedback ◽  
Space Representation ◽  
Control Networks ◽  
State Space Representation ◽  
Robust Synchronization ◽  
Necessary And Sufficient Criteria ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Product Of Matrices ◽  
Theoretical Results

This paper investigates the robust synchronization of drive-response Boolean control networks (BCNs) with disturbances via semi-tensor product of matrices. Firstly, the definition of robust synchronization is presented for the drive-response BCNs with disturbances. Then, based on the algebraic state space representation of drive-response BCNs, the robustly reachable states/sets are presented to investigate robust synchronization of disturbed BCNs. According to the set of robustly reachable states, some necessary and sufficient criteria are obtained for robust synchronization of drive-response BCNs with disturbances under a given state feedback controller. Finally, an illustrative example is presented to demonstrate the obtained theoretical results.

scholarly journals Matrix Formulation of EISs of Graphs and Its Application to WSN Covering Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yongyi Yan ◽  
Jumei Yue ◽  
He Deng
Keyword(s):  
Sufficient Conditions ◽  
Independent Sets ◽  
Complete Characterization ◽  
Covering Problem ◽  
Matrix Formulation ◽  
Covering Problems ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
Product Of Matrices

In this paper, the problem of formulating and finding externally independent sets of graphs is considered by using a newly developed STP method, called semitensor product of matrices. By introducing a characteristic value of a vertex subset of a graph and using the algebraic representation of pseudological functions, several necessary and sufficient conditions of matrix form are proposed to express the externally independent sets (EISs), minimum externally independent sets (MEISs), and kernels of graphs. Based on this, the concepts of EIS matrix, MEIS matrix, and kernel matrix are introduced. By these matrices’ complete characterization of these three structures of graphs, three algorithms are further designed which can find all these kinds of subsets of graphs mathematically. The results are finally applied to a WSN covering problem to demonstrate the correctness and effectiveness of the proposed results.

Necessary and Sufficient Conditions for Group Consensus of Agents With Third-Order Dynamics in Directed Networks

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Onur Cihan ◽  
Mehmet Akar
Keyword(s):  
Directed Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Consensus ◽  
Directed Networks ◽  
Third Order ◽  
Systematic Method ◽  
Necessary And Sufficient ◽  
The Given ◽  
Theoretical Results

Abstract In this paper, we investigate the group consensus problem in directed networks where agents have third-order dynamics. Necessary and sufficient conditions on the controller parameters are obtained to ensure K-equilibria group consensus where K is determined by the structure of the directed graph. It is theoretically shown that, for an arbitrary directed graph, there exist controller parameters that satisfy the given conditions. A systematic method for choosing the controller parameters to guarantee group consensus is suggested and theoretical results are verified by numerical examples.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

A DNA computer model for solving vertex coloring problem

2006 ◽  
Vol 51 (20) ◽  
pp. 2541-2549 ◽  
Author(s):  
Jin Xu ◽  
Xiaoli Qiang ◽  
Fang Gang ◽  
Kang Zhou
Keyword(s):  
Computer Model ◽  
Vertex Coloring ◽  
Coloring Problem ◽  
Vertex Coloring Problem ◽  
Dna Computer

scholarly journals Optimising the AR Engraved Structure on Light-Guide Facets for a Wide Range of Wavelengths

Optics ◽  
2020 ◽  
Vol 2 (1) ◽  
pp. 25-42
Author(s):  
Ioseph Gurwich ◽  
Yakov Greenberg ◽  
Kobi Harush ◽  
Yarden Tzabari
Keyword(s):  
Theoretical Analysis ◽  
Light Guide ◽  
Spectral Band ◽  
Angular Range ◽  
The Past ◽  
Input And Output ◽  
Wide Range ◽  
The Given ◽  
Theoretical Results ◽  
Shape And Size

The present study is aimed at designing anti-reflective (AR) engraving on the input–output surfaces of a rectangular light-guide. We estimate AR efficiency, by the transmittance level in the angular range, determined by the light-guide. Using nano-engraving, we achieve a uniform high transmission over a wide range of wavelengths. In the past, we used smoothed conical pins or indentations on the faces of light-guide crystal as the engraved structure. Here, we widen the class of pins under consideration, following the physical model developed in the previous paper. We analyze the smoothed pyramidal pins with different base shapes. The possible effect of randomization of the pins parameters is also examined. The results obtained demonstrate optimized engraved structure with parameters depending on the required spectral range and facet format. The predicted level of transmittance is close to 99%, and its flatness (estimated by the standard deviation) in the required wavelengths range is 0.2%. The theoretical analysis and numerical calculations indicate that the obtained results demonstrate the best transmission (reflection) we can expect for a facet with the given shape and size for the required spectral band. The approach is equally useful for any other form and of the facet. We also discuss a simple way of comparing experimental and theoretical results for a light-guide with the designed input and output features. In this study, as well as in our previous work, we restrict ourselves to rectangular facets. We also consider the limitations on maximal transmission produced by the size and shape of the light-guide facets. The theoretical analysis is performed for an infinite structure and serves as an upper bound on the transmittance for smaller-size apertures.

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