scholarly journals Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1522
Author(s):  
Anna Concas ◽  
Lothar Reichel ◽  
Giuseppe Rodriguez ◽  
Yunzi Zhang
Keyword(s):  
Iterative Methods ◽  
Adjacency Matrix ◽  
Lanczos Method ◽  
Power Method ◽  
Arnoldi Method ◽  
Multiple Eigenvalues ◽  
Computer Storage ◽  
Adjacency Matrices ◽  
Lanczos Methods ◽  
Perron Vector

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

scholarly journals COMB GRAPHS AND SPECTRAL DECIMATION

2009 ◽  
Vol 51 (1) ◽  
pp. 71-81 ◽  
Author(s):  
JONATHAN JORDAN
Keyword(s):  
Adjacency Matrix ◽  
Spectral Properties ◽  
Sierpinski Gasket ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Spectral Similarity ◽  
Similarity Relationship ◽  
Adjacency Matrices ◽  
Similarity Operators ◽  
Math Phys

AbstractWe investigate the spectral properties of matrices associated with comb graphs. We show that the adjacency matrices and adjacency matrix Laplacians of the sequences of graphs show a spectral similarity relationship in the sense of work by L. Malozemov and A. Teplyaev (Self-similarity, operators and dynamics,Math. Phys. Anal. Geometry6(2003), 201–218), and hence these sequences of graphs show a spectral decimation property similar to that of the Laplacians of the Sierpiński gasket graph and other fractal graphs.

scholarly journals aMatReader: Importing adjacency matrices via Cytoscape Automation

F1000Research ◽  
2018 ◽  
Vol 7 ◽  
pp. 823
Author(s):  
Brett Settle ◽  
David Otasek ◽  
John H Morris ◽  
Barry Demchak
Keyword(s):  
Adjacency Matrix ◽  
Pairwise Interaction ◽  
Interaction Data ◽  
Gene Pairs ◽  
Link Type ◽  
Analysis Tools ◽  
Adjacency Matrices

Adjacency matrices are useful for storing pairwise interaction data, such as correlations between gene pairs in a pathway or similarities between genes and conditions. The aMatReader app enables users to import one or multiple adjacency matrix files into Cytoscape, where each file represents an edge attribute in a network. Our goal was to import the diverse adjacency matrix formats produced by existing scripts and libraries written in R, MATLAB, and Python, and facilitate importing that data into Cytoscape. To accelerate the import process, aMatReader attempts to predict matrix import parameters by analyzing the first two lines of the file. We also exposed CyREST endpoints to allow researchers to import network matrix data directly into Cytoscape from their language of choice. Many analysis tools deal with networks in the form of an adjacency matrix, and exposing the aMatReader API to automation users enables scripts to transfer those networks directly into Cytoscape with little effort.

scholarly journals The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Fatih Yılmaz ◽  
Durmuş Bozkurt
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Intensive Study ◽  
Integer Power ◽  
Determinantal Representation ◽  
Adjacency Matrices

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the(i,j)entry ofAm(Ais adjacency matrix) is equal to the number of walks of lengthmfrom vertexito vertexj, we show that elements ofmth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

Block Adjacency Matrix Method for Analyzing the Configuration Transformations of Metamorphic Parallel Mechanisms

2014 ◽  
Author(s):  
Duanling Li ◽  
Chunxia Li ◽  
Zhonghai Zhang ◽  
Xianwen Kong
Keyword(s):  
Adjacency Matrix ◽  
Matrix Method ◽  
Parallel Mechanism ◽  
New Method ◽  
Parallel Mechanisms ◽  
Adjacency Matrices ◽  
Moving Platform ◽  
Internal Configuration ◽  
Spherical Motion ◽  
Metamorphic Mechanisms

Metamorphic transformation is a fundamental and key issue in the design and analysis of metamorphic mechanisms. It is tedious to represent and calculate the metamorphic transformations of metamorphic parallel mechanisms using the existing adjacency matrix method. To simplify the configuration transformation analysis, we propose a new method based on block adjacency matrix to analyze the configuration transformations of metamorphic parallel mechanisms. A block adjacency matrix is composed of three types of elements, including limb matrices that are adjacency matrices each representing a limb of a metamorphic parallel mechanism, row matrices each representing how a limb is connected to the moving platform, and column matrices each representing how a limb is connected to the base. Manipulations of the block adjacency matrix for analyzing the metamorphic transformations are presented systematically. If only the internal configuration of a limb changes, the configuration transformations can be obtained by simply calculating the corresponding limb matrix. A 3-URRRR metamorphic parallel mechanism, which has five configurations including a 1-DOF translation configuration and a 3-DOF spherical motion configuration, is taken as an example to illustrate the effectiveness of the proposed approach to the metamorphic transformation analysis of metamorphic parallel mechanism.

Conceptual design of the hybrid hydraulic system based on adjacency matrices operations

2017 ◽  
Vol 232 (7) ◽  
pp. 1179-1190
Author(s):  
Ming Xiang ◽  
Delun Wang
Keyword(s):  
Physical Model ◽  
Conceptual Design ◽  
Design Process ◽  
Adjacency Matrix ◽  
Hydraulic System ◽  
Control Valves ◽  
Directional Control ◽  
Adjacency Matrices ◽  
Design Requirements ◽  
Distribution Of Flow

This paper presents a new method to describe the selection design of directional control valves in the form of matrix. The directional control valves in hybrid are the basic units to implement the distribution of flow in the hydraulic system. Both the design requirements of the hybrid hydraulic system and the basic units are expressed by the adjacency matrices. Therefore, the selection design of directional control valves is the process to decompose the adjacency matrix of the system into a series of sub-matrices according to the decomposition rules. The defined rules of adjacency matrices are obtained according with the composite physical model of several valves. For obtaining easily the selection design of directional control valves, the library of basic units is established. The style of directional control valves is obtained by matching the sub-matrices with the matrices of basic units. Through configuring the others components, a thorough conceptual design process of the hydraulic system is established. Example is given to illustrate the whole design process in detail.

scholarly journals Alpha Adjacency: A generalization of adjacency matrices

2019 ◽  
Vol 35 ◽  
pp. 365-375
Author(s):  
Matt Hudelson ◽  
Judi McDonald ◽  
Enzo Wendler
Keyword(s):  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Average Characteristic ◽  
Adjacency Matrices ◽  
Skew Adjacency Matrix

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

Adjacency matrix comparison for stochastic block models

2019 ◽  
Vol 08 (03) ◽  
pp. 1950010
Author(s):  
Guangren Yang ◽  
Songshan Yang ◽  
Wang Zhou
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Adjacency Matrix ◽  
Real Data ◽  
Finite Sample ◽  
Adjacency Matrices ◽  
Block Models

In this paper, we study whether two networks arising from two stochastic block models have the same connection structures by comparing their adjacency matrices. We conduct Monte Carlo simulations study to examine the finite sample performance of the proposed method. A real data example is used to illustrate the proposed methodology.

scholarly journals Indefinite Ruhe’s Variant of the Block Lanczos Method for Solving the Systems of Linear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
M. Ghasemi Kamalvand ◽  
K. Niazi Asil
Keyword(s):  
Scalar Product ◽  
Hermitian Matrix ◽  
Linear Equations ◽  
Lanczos Method ◽  
Krylov Methods ◽  
Block Arnoldi ◽  
Systems Of Linear Equations ◽  
Block Lanczos ◽  
Lanczos Methods ◽  
Block Lanczos Method

In this paper, we equip Cn with an indefinite scalar product with a specific Hermitian matrix, and our aim is to develop some block Krylov methods to indefinite mode. In fact, by considering the block Arnoldi, block FOM, and block Lanczos methods, we design the indefinite structures of these block Krylov methods; along with some obtained results, we offer the application of this methods in solving linear systems, and as the testifiers, we design numerical examples.

On the Aα-spectrum of joined union of digraphs

2021 ◽  
pp. 2150086
Author(s):  
Hilal A. Ganie
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Adjacency Matrices ◽  
Matrix Formula ◽  
Regular Digraph

Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.

Isomorphism Identification of Graphs of Kinematic Chains

2007 ◽  
Author(s):  
Huafeng Ding ◽  
Zhen Huang
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Adjacency Matrix ◽  
Complexity Analysis ◽  
Unique Representation ◽  
Kinematic Chains ◽  
Adjacency Matrices ◽  
Isomorphism Identification

Isomorphism identification of graphs is one of the most important and challenging problems in the fields of mathematics, computer science and mechanisms. This paper attempts to solve the problem by finding a unique representation of graphs. First, the perimeter loop of a graph is identified from all the loops of the graph obtained through a new algorithm. From the perimeter loop a corresponding perimeter graph is derived, which renders the forms of the graph canonical. Then, by relabelling the perimeter graph, the canonical perimeter graph is obtained, reducing the adjacency matrices of a graph from hundreds of thousands to several or even just one. On the basis of canonical adjacency matrix set, the unique representation of the graph, the characteristic adjacency matrix, is obtained. In such a way, isomorphism identification, sketching, and establishment of the database of common graphs, including the graphs of kinematic chains, all become easy to realize. Computational complexity analysis shows that, in the field of kinematic chains the approach is much more efficient than McKay’s algorithm which is considered the fastest so far. Our algorithm remains efficient even when the links of kinematic chains increase into the thirties.

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