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In this paper a new method is proposed for finite element domain decomposition. A weighted incidence graph is first constructed for the finite element model. A spectral partitioning heuristic is then applied to the graph using the second and the third eigenvalues of the Laplacian matrix of the graph, to partition it into three subgraphs and correspondingly trisect the finite element model.

Nodal ordering for the formation of well‐structures stiffness matrices are often performed using graph theory and algebraic graph theory. The purpose of this paper is to present a new method for nodal ordering for profile optimization of finite element models.

In the present method, a combination of graph theory and differential equations is employed. The proposed method transforms the eigenvalue problem involved in optimal ordering of algebraic graph method into a specific initial value problem of an ordinary differential equation.

The transformation of this paper enables many advanced numerical methods for ordinary differential equations to be used in the computation of the eigenproblems.

Combining two different tools, namely graph theory and differential equations, results in a more efficient and accurate method for nodal ordering problem, which is a combinatorial optimization problem. Examples are included to illustrate the efficiency of the present method.

Domain decomposition of finite element models (FEM) for parallel computing are often performed using graph theory and algebraic graph theory. This paper aims to present a new method for such decomposition, where a combination of algebraic graph theory and differential equations is employed.

In the present method, a combination of graph theory and differential equations is employed. The proposed method transforms the eigenvalue problem involved in decomposing FEM by the algebraic graph method, into a specific initial value problem of an ordinary differential equation.

The transformation of this paper enables many advanced numerical methods for ordinary differential equations to be used in the computation of the eigenproblems.

Combining two different tools, namely algebraic graph theory and differential equations, results in an efficient and accurate method for decomposing the FEM which is a combinatorial optimization problem. Examples are included to illustrate the efficiency of the present method.

For the solution of equations with sparse matrices, the problem of bandwidth reduction is an important issue. Though graph theoretical algorithms are available, the purpose of this paper is to examine the feasibility of ant systems (AS).

For band optimization an ant colony algorithm based on AS is utilized. In this algorithm a local search procedure is also included to improve the solution.

AS algorithms are found to be suitable for bandwidth optimization.

Application of AS to the bandwidth reduction is the main purpose of this paper, which is successfully performed. The results are compared to those of a graph theoretical bandwidth optimization algorithm.

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.

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.

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

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.