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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 88
Edited by: B.H.V. Topping and M. Papadrakakis
Paper 166

Interrelation of Group Products and Graph Products in Configuration Processing of Symmetric Structures

A. Kaveh and M. Nikbakht

Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

Full Bibliographic Reference for this paper
A. Kaveh, M. Nikbakht, "Interrelation of Group Products and Graph Products in Configuration Processing of Symmetric Structures", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 166, 2008. doi:10.4203/ccp.88.166
Keywords: graph products, group product, composition, generator, direct product, Kronecker product.

Graph theory has a long history, and its applications in structural mechanics and in particular nodal ordering and graph partitioning are well documented in the literature, Kaveh [1,2]. Group theory has also found applications in structural mechanics. See for example Zingoni [3,4].

Many structural models can be generated as the graph products of two or three subgraphs known as their generators. Graph products have applications in structural mechanics, Kaveh and Rahami [5].

In this paper, after a brief review on basic concepts of group theory and properties of the symmetry groups, different products which are defined in graph theory are introduced. Kronecker product of the matrices and direct product of the groups are then defined and the inter-relationship between the graph products and the direct product of the groups will be investigated in symmetric graphs with symmetric or asymmetric generators. This paper highlights the physical interpretation of these two products which are associated with the same algebraic base. Graph products have been well-established and implemented recently in structural mechanics and can help to improve the concept of group product, or be improved in accordance with this concept.

Configuration analysis of special symmetric systems which can be expressed as the product of two generators is studied in this paper. In such structures, symmetrical properties of the system can be predicted from the properties of the generators, by the means of group theory. Such configurations are used extensively in civil engineering, because it enables the designers to create superstructures showing complicated forms of symmetry, which are not only pleasant from the architectural aspects, but also show ideal structural performances, using the product of very simple generators.

The method presented in this paper provides a simple technique to find these complex symmetrical properties of the graph products from the symmetry elements of their generators. Basic concepts in group theory are implemented in this regard and the abilities and limitations of this tool are studied. It is shown that when two symmetric graphs are composed to each other by the means of the graph product, and generate a composed graph, in most cases new symmetrical properties are produced. The direct product of symmetry groups of the generators yields a symmetry group which is isomorphic to the symmetry group of the composed graph, or at least to a subgroup of it. Since all of the symmetrical properties of a symmetric system are extractable from its symmetry group, it will be advantageous to be able to predict the high-ordered symmetry group of the complicated composed system.

Kaveh A., "Structural Mechanics: Graph and Matrix Methods", Research Studies Press, 3rd ed, UK, 2004.
Kaveh A., "Optimal Structural Analysis", Research Studies Press, 2nd edition, John Wiley, UK, 2006.
Zingoni A., "An efficient computational scheme for the vibration analysis of high-tension cable nets, Journal of Sound and Vibration, No. 1, 189, 55-79, 1996. doi:10.1006/jsvi.1996.0005
Zingoni A., "Group-theoretical applications in solid and structural mechanics: a review", Chapter 12, Computational Structures Technology, Eds: BHV Topping & Z. Bittnar, Saxe-Coburg Publications, UK, 2002. doi:10.4203/csets.7.12
Kaveh A., Rahami H., "Block diagonalization of adjacency and Laplacian matrices for graph products; Applications in structural mechanics", International Journal for Numerical Methods in Engineering, No. 1, 68, 33-63, 2006. doi:10.1002/nme.1696

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