Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products

SUMMARYTwo important matrices associated with graphs are adjacency and Laplacian matrices. In this paper efficient methods are presented for eigensolution of graph products augmented by other graphs. For augmentations that do not destroy the symmetry of the graph products, a method is proposed for decomposition of matrices resulting in considerable simplification of their eigensolution. For graphs composed of two non-overlapping graph products joined through a small number of link members, a method based on shifted inverse iteration is proposed which utilizes all eigenvalues and eigenvectors of each individual graph products. Owing to the availability of fast methods for eigensolution of graph products, this method simplifies the eigensolution of a variety of graph models and proves to be very efficient in determining the few smallest eigenpairs of these models with high levels of accuracy.

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“…Let v be the normalized vector of order n, with all the entries being equal to unity, as defined in Equation (9). It is clear that…”

Section: The Choice Of Shifts and Starting Vectorsmentioning

confidence: 99%

Eigensolution of augmented graph products using shifted inverse iteration method

Kaveh

Fazli

2010

Numerical Meth Engineering

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“…depending on the expanding generator being a circle or a path, respectively (Kaveh, 2006). Repetitive structures have been widely studied by Kangwai et al (1999), Kangwai and Guest (2000), Kaveh and Rahami (2004, 2007, 2008, Kaveh et al (2012Kaveh et al ( , 2013, , Koohestani (2010), and Zingoni (2009Zingoni ( , 2012aZingoni ( , 2012bZingoni ( , 2014. Kaveh et al (2013) analyzed the near-regular structure by using the force method.…”

Section: Equivalent Circulant Structures 343mentioning

confidence: 99%

“…Repetitive structures can be divided into two categories, namely, circulant and regular structures, depending on the expanding generator being a circle or a path, respectively (Kaveh, 2006). Repetitive structures have been widely studied by Kangwai et al (1999), Kangwai and Guest (2000), Kaveh and Rahami (2004, 2007, 2008), Kaveh et al (2012, 2013), Rahami (2012), Koohestani (2010), and Zingoni (2009, 2012a, 2012b, 2014).…”

Section: Introductionmentioning

confidence: 99%

Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Kaveh

Rahami

Jodaki

2017

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Purpose There are many structures that have a repetitive pattern. If a relationship can be established between a repetitive structure and a circulant structure, then the repetitive structure can be analyzed by using the properties of the corresponding circulant structure. The purpose of this paper is to develop such a transformation. Design/methodology/approach A circulant matrix has certain properties that can be used to reduce the complexity of the analysis. In this paper, repetitive and near-repetitive structures are transformed to circulant structures by adding and/or eliminating some elements of the structure. Numerical examples are provided to show the efficiency of the present method. Findings A transformation is established between a repetitive structure and a circulant structure, and the analysis of the repetitive structure is performed by using the properties of the corresponding circulant structure. Originality/value Repetitive and near-repetitive structures are transformed to circulant structures, and the complexity of the analysis of the former structures is reduced by analyzing the latter structures.

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“…These eigenvalues play a central role in our fundamental understanding of graphs.the corresponding canonical forms. Thus, the eigenvalues of the entire graph can be obtained as the union of the eigenvalues of much smaller submatrices [13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of the adjacency and Laplacian matrices for modified regular structural models

Kaveh

Rahami

2009

Numer Methods Biomed Eng

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SUMMARYThe graph model of many space structures and finite element models can be formed using graph products. These structures are known as regular structures. Methods for calculating the eigenvalues of the matrices corresponding to these models are already investigated. In practice for some structural models the addition of some nodes and members to the graph product is required. When such a node and the incident members are added to a regular model, it becomes a modified regular structure. In this paper the eigenproblem of the matrices corresponding to these models is studied. The necessary formulations are derived and examples are presented to illustrate the practical value of the presented approach.

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Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products

Cited by 21 publications

References 17 publications

Eigensolution of augmented graph products using shifted inverse iteration method

Eigensolution of augmented graph products using shifted inverse iteration method

Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Eigenvalues of the adjacency and Laplacian matrices for modified regular structural models

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