Laplacian Matrices (laplacian + matrix)

Distribution by Scientific Domains


Selected Abstracts


Additivity properties of graphs with Form II symmetry

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2006
A. Kaveh
Abstract In this paper, the properties of previously developed Form II symmetry (Commun. Numer. Meth. Engng 2003; 19:125,136; 2004; 20:133,146) is further investigated. Additivity properties of graphs with this form are formulated, and the effect of adding or deleting members on the eigenvalues of the Laplacian matrices of the corresponding graphs, is studied. Depending on the category of the added or deleted members, the condensed submatrices on which changes occur are identified, and the necessary modifications are suggested. A mass-spring dynamic system is presented to illustrate a typical application of the latter approach. Copyright © 2005 John Wiley & Sons, Ltd. [source]


A unified method for eigendecomposition of graph products

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2005
A. Kaveh
Abstract In this paper, a unified method is developed for calculating the eigenvalues of the weighted adjacency and Laplacian matrices of three different graph products. These products have many applications in computational mechanics, such as ordering, graph partitioning, and subdomaining of finite element models. Copyright © 2005 John Wiley & Sons, Ltd. [source]


Block diagonalization of Laplacian matrices of symmetric graphs via group theory

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2007
A. Kaveh
Abstract In this article, group theory is employed for block diagonalization of Laplacian matrices of symmetric graphs. The inter-relation between group diagonalization methods and algebraic-graph methods developed in recent years are established. Efficient methods are presented for calculating the eigenvalues and eigenvectors of matrices having canonical patterns. This is achieved by using concepts from group theory, linear algebra, and graph theory. These methods, which can be viewed as extensions to the previously developed approaches, are illustrated by applying to the eigensolution of the Laplacian matrices of symmetric graphs. The methods of this paper can be applied to combinatorial optimization problems such as nodal and element ordering and graph partitioning by calculating the second eigenvalue for the Laplacian matrices of the models and the formation of their Fiedler vectors. Considering the graphs as the topological models of skeletal structures, the present methods become applicable to the calculation of the buckling loads and the natural frequencies and natural modes of skeletal structures. Copyright © 2006 John Wiley & Sons, Ltd. [source]


An efficient method for decomposition of regular structures using graph products

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2004
A. Kaveh
Abstract In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd. [source]