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Eigensolution of Laplacian matrices for graph partitioning and domain decomposition: approximate algebraic method. (English) Zbl 1257.65020

Summary: Purpose: The purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.

Design/methodology/approach: In the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.

Findings: A new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.

Originality/value: The present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

MSC:
65F15Eigenvalues, eigenvectors (numerical linear algebra)
05C90Applications of graph theory