scholarly journals Fast Algorithm for Calculating Transitive Closures of Binary Relations in the Structure of a Network Object

2021 ◽  
pp. 560-566
Author(s):  
Vladimir Batsamut ◽  
Sviatoslav Manzura ◽  
Oleksandr Kosiak ◽  
Viacheslav Garmash ◽  
Dmytro Kukharets
Keyword(s):  
Adjacency Matrix ◽  
Fast Algorithm ◽  
Execution Time ◽  
Control Algorithms ◽  
Network Structures ◽  
Binary Relations ◽  
Transmission Networks ◽  
The Matrix ◽  
Different Dimensions ◽  
Network Object

The article proposes a fast algorithm for constructing the transitive closures between all pairs of nodes in the structure of a network object, which can have both directional and non-directional links. The algorithm is based on the disjunctive addition of the elements of certain rows of the adjacency matrix, which models (describe) the structure of the original network object. The article formulates and proves a theorem that using such a procedure, the matrix of transitive closures of a network object can be obtained from the adjacency matrix in two iterations (traversal) on such an array. An estimate of the asymptotic computational complexity of the proposed algorithm is substantiated. The article presents the results of an experimental study of the execution time of such an algorithm on network structures of different dimensions and with different connection densities. For this indicator, the developed algorithm is compared with the well-known approaches of Bellman, Warshall-Floyd, Shimbel, which can also be used to determine the transitive closures of binary relations of network objects. The corresponding graphs of the obtained dependences are given. The proposed algorithm (the logic embedded in it) can become the basis for solving problems of monitoring the connectivity of various subscribers in data transmission networks in real time when managing the load in such networks, where the time spent on routing information flows directly depends on the execution time of control algorithms, as well as when solving other problems on the network structures.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals HILL CIPHER ON SHAMIR’S THREE PASS PROTOCOL

2017 ◽  
Author(s):  
hasdiana
Keyword(s):  
Matrix Element ◽  
Execution Time ◽  
Invertible Matrix ◽  
Multidisciplinary Research ◽  
The Third ◽  
Xor Operation ◽  
Hill Cipher ◽  
The Matrix ◽  
Level Of Difficulty ◽  
Time Required

This preprint has been presented in the 3rd International Conference on Multidisciplinary Research, Medan, october 16 – 18, 2014---In this study the authors use the scheme of Shamir's Three Pass Protocol for Hill Cipher operation. Scheme of Shamir's Three Pass Protocol is an attractive scheme that allows senders and receivers to communicate without the key exchange. Hill Cipher is chosen because of the key-shaped matrix, which is expected to complicate the various techniques of cryptanalyst. The results of this study indicate that the weakness of the scheme of Shamir's Three Pass Protocol for XOR operation is not fully valid if it is used for Hill Cipher operations. Cryptanalyst can utilize only the third ciphertext that invertible. Matrix transpose techniques in the ciphertext aims to difficulties in solving this algorithm. The original ciphertext generated in each process is different from the transmitted ciphertext. The level of difficulty increases due to the use of larger key matrix. The amount of time required for the execution of the program depends on the length of the plaintext and the value of the matrix element. Plaintext has the same length produce different execution time depending on the value of the key elements of the matrix used.

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

A Variational Approach for Modeling Flexibility Effects in Manipulator Arms

Robotica ◽  
1991 ◽  
Vol 9 (2) ◽  
pp. 213-217 ◽  
Author(s):  
Ali Meghdari
Keyword(s):  
Motion Control ◽  
Variational Approach ◽  
Control Algorithms ◽  
General Technique ◽  
Robotic Arms ◽  
Cartesian Space ◽  
The Matrix ◽  
Structural Deformations ◽  
Castigliano’S Theorem ◽  
Compliance Coefficients

SUMMARYThis paper presents a general technique to model flexible components (mainly links and joints flexibilities are considered) of manipulator arms based on Castigliano's theorem of least work. The robotic arms flexibility properties are derived and represented by the matrix of compliance coefficients. Such expressions can be used to determine the errors due to the robotic tip deformations under the application of a set of applied loads at the tip in a Cartesian space. Once these deformations are computed, they may be used to correct for the positional errors arisen from the robotic structural deformations in the motion control algorithms.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

Performance Analysis of the PISO-Based CFD Simulation

2014 ◽  
Vol 607 ◽  
pp. 872-876 ◽  
Author(s):  
Xiao Guang Ren
Keyword(s):  
Performance Optimization ◽  
Execution Time ◽  
Cfd Simulation ◽  
Fluid Flows ◽  
Total Execution Time ◽  
Simulation Performance ◽  
Simulation Efficiency ◽  
Matrix Operations ◽  
The Matrix ◽  
Computational Fluid Dynamics Cfd

Computational Fluid Dynamics (CFD) is widely applied for the simulation of fluid flows, and the performance of the simulation process is critical for the simulation efficiency. In this paper, we analyze the performance of CFD simulation application with profiling technology, which gets the portions of the main parts’ execution time. Through the experiment, we find that the PISO algorithm has a significant impact on the CFD simulation performance, which account for more than 90% of the total execution time. The matrix operations are also account for more than 60% of the total execution time, which provides opportunity for performance optimization.

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Charles Delorme ◽  
Guillermo Pineda-Villavicencio
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Minimum Degree ◽  
Integer Coefficients ◽  
Disjoint Cycles ◽  
The Matrix ◽  
Odd Degree ◽  
Unexplored Area ◽  
Trivial Graph ◽  
Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

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