Block diagonalization of Laplacian matrices of symmetric graphs via group theory

Kaveh, A.; Nikbakht, M.

doi:10.1002/nme.1794

Cited by 24 publications

(9 citation statements)

References 21 publications

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“…In this section, we describe ways of scaling this approach to large discrete spaces. We investigate several approaches, ranging from exploiting the structure of highly symmetric graphs [32,63], to the use of sparsification and sampling methods [39] to streamline matrix computations. We first analyze structured graphs that are constructed from simpler graphs, based on the notion of a Kronecker product [32].…”

Section: Scaling To Large State Space Graphsmentioning

confidence: 99%

Learning Representation and Control in Markov Decision Processes: New Frontiers

Mahadevan

2007

FNT in Machine Learning

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Section: Scaling To Large State Space Graphsmentioning

confidence: 99%

Learning Representation and Control in Markov Decision Processes: New Frontiers

Mahadevan

2007

FNT in Machine Learning

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“…If the mass associated with that node moves in the positive direction, we usually use the notation (towards the plane) and if the mass moves in the negative direction we use the notation (out of the plane) for that node. Such a notation has been previously utilized by the authors [20], which is inspired from the works of Zingoni [4,17]. Comparing the correspondence between the stiffness matrix of the mass-spring system and the Laplacian matrix of its graph model, one can obviously show the similarity of physical properties of the mass-spring system and its graph model.…”

Section: Canonical Form IImentioning

confidence: 99%

Decomposition of symmetric mass–spring vibrating systems using groups, graphs and linear algebra

Kaveh

Nikbakht

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe main objective of this article is to develop a methodology for an efficient calculation of the eigenvalues for symmetric mass-spring systems in order to reduce the size of the eigenproblem involved. This is achieved using group-theoretical method, whereby the model of a symmetric mass-spring system is decomposed into appropriate submodels. The eigenvalues of the entire system is then obtained by calculating the eigenvalues of its submodels. The results are compared to those of the existing methods based on graph theory and linear algebra. Examples are provided to illustrate the simplicity and efficiency of the present method.

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“…EXPLOITATION OF SYMMETRY IN GRAPHS 153 element stiffness and consistent mass matrices for different symmetric finite elements. Block diagonalization of Laplacian matrices of symmetric graphs was presented by Kaveh and Nikbakht [25]. Group theory was also used for efficient topology optimization of truss structures by Bai et al [26].…”

Section: Introductionmentioning

confidence: 99%

Exploitation of symmetry in graphs with applications to finite and boundary elements analysis

Koohestani

2012

Numerical Meth Engineering

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We present explicit and parametric forms of transformation matrices for three well-known and widely used symmetry groups: S 2 , C 2v and C 4v . Group representation theory is the most powerful method for exploiting symmetry. We propose an efficient algorithm for systematic generation of reducible representations that can be combined linearly to obtain the projection operators. The exact column spaces of these projection operators are calculated and integrated through special orderings, leading to exact explicit and parametric forms of transformation matrices. The transformation matrices could be used directly for block diagonalization of single-variable scalar field problems. Another algorithm is proposed to extend the application of the method to nonscalar and multivariable field problems. Finally, the generality and efficiency of the proposed method in relation to computation times and the accuracy of results are illustrated through examples from spectral decomposition, free vibration, buckling of FEMs and boundary element analysis of a symmetric field. One of the earliest applications of group representation theory to the stability of structures was developed by Renton [5]. The theory has since been used extensively for the study of local and global bifurcation analyses. Among the outstanding studies in bifurcation analysis are Sattinger [6], Golubitsky and Schaeffer [7,8], Ikeda and Murota [9, 10], Ikeda et al. [11,12], Healey [13] and Wohlever and Healey [14]. Group representation theory has also been employed effectively in the vibration analysis of symmetric structures by Healey and Treacy [15], Zlokovic [16], Zingoni [17, 18] and Mohan and Pratap [19]. Eigenvalue problems related to the vibration of massspring systems were studied by Kaveh and Nikbakht [20], Kaveh and Jahanmohammadi [21] and Zingoni [22] using group theory. Zingoni [23,24] used group theory to simplify the generation of Node adjacency matrix: Let G be a graph with N nodes. The adjacency matrix A is an N N matrix in which the entry in row i and column j is 1 if node i is adjacent toj , and is 0 otherwise. Note that, for simple graphs (graphs without loops and multiple edges) the diagonal entries of the adjacency matrix are always zero; however, the existence of k loops at node i implies that node i is connected to itself by k edges. In this case, the i-th diagonal entry in the adjacency matrix should equal k.Laplacian matrix: The Laplacian matrix of graph G (denoted by L) is defined as Symmetry subspacesBy defining a coordinate system for a configuration (the configuration could be a graph, a structural model, etc.) we can represent all symmetry operations as matrices. These matrices will change according to different forms of coordinate systems. This form of representation is called the reducible matrix representation and denoted here by P(see [29] for more details). However, there is always a special coordinate system (a symmetry-adapted coordinate system) such that each symmetry operation is represented on it as a unique matrix. These f...

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Block diagonalization of Laplacian matrices of symmetric graphs via group theory

Cited by 24 publications

References 21 publications

Learning Representation and Control in Markov Decision Processes: New Frontiers

Learning Representation and Control in Markov Decision Processes: New Frontiers

Decomposition of symmetric mass–spring vibrating systems using groups, graphs and linear algebra

Exploitation of symmetry in graphs with applications to finite and boundary elements analysis

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