Eigensolution of Laplacian matrices for graph partitioning and domain decomposition: Approximate algebraic method

Abstract

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.

Citation

Kaveh, A. and Alinejad, B. (2009), "Eigensolution of Laplacian matrices for graph partitioning and domain decomposition: Approximate algebraic method", Engineering Computations, Vol. 26 No. 7, pp. 828-842. https://doi.org/10.1108/02644400910985198