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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 95
Topological and Graph Products: Eigenproblems for Optimal Structural Analysis A. Kaveh and H. Rahami
Iran University of Science and Technology, Narmak, Tehran, Iran Full Bibliographic Reference for this paper
A. Kaveh, H. Rahami, "Topological and Graph Products: Eigenproblems for Optimal Structural Analysis", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 95, 2005. doi:10.4203/ccp.81.95
Keywords: graph product, Cartesian product, strong Cartesian product, direct product, eigenproblem, regular graph.
Summary
In this paper, a topological method is presented to obtain the eigensolution of a
cylindrical-shaped grid from those of a plane grid. Similarly, the properties of a
torus-shaphed grid is calculated from those of a cylindrical-shaped grid. This
method simplifies the evaluation of the Fiedler vectors to be employed in the
ordering and graph partitioning of regular structural models, for optimal structural
analysis.
Consider a planar grid
In this paper it is proved that
As an example, for
Similarly, it can be proved that the third eigenvalue
For direct and strong Cartesian products the same results are applicable, and
As an example, for
The eigenvalues of the adjacency and Laplacian matrices of three different graph products, provide an efficient approach for calculating the eigenvalues of adjacency and Laplacian matrices of weighted and non-weighted graphs. The eigensolution of graphs has many applications in computational mechanics. Examples of such applications are nodal and element ordering for bandwidth, profile and frontwidth optimization, graph partitioning, and subdomaining of finite element models [1,2]. The present forms are also effective tools for calculating eigenvalues and eigenvectors of matrices arising form numerical methods for differential equations applied to structural mechanics problems. References
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