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©2012 Civil-Comp Ltd |
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A. Kaveh and B. Alinejad
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran
Keywords: hypergraph products, dynamic graph products, variable adjacency function, configuration processing, adjacency matrix, Laplacian matrix.
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Graph products have been studied extensively in the last decade and applied to many problems in structural mechanics, including configuration processing, parallel computing, and optimal analysis of structures. Hypergraphs are generalized forms of graphs and many applications of graphs can be extended to hypergraphs because of their generality. In a hypergraph there are vertices and edges and there is no distinct definition for adjacency. To tackle this problem adjacency functions are defined for hypergraphs and hence distinct adjacency and Laplacian matrices are achieved. Different adjacency functions can be defined for a given problem and therefore the hypergraphs can be applied to a broader domain of problems compared to graphs.
In structural mechanics the behaviour of the system may vary with respect to the time or any other variable, the under study system will be dynamic. Solving the dynamic problems are more involved than static ones and therefore any simplification in the process of their solution will be valuable. Hypergraph products with variable adjacency functions for their sub-hypergraphs, results in variable graph products that are called dynamic graph products. From the application point of view, the dynamic graph products can be used in modelling and configuration processing of dynamic regular structures. Furthermore using the related algebraic attributes of their matrices leads to further simplification of their solution procedures.
In this paper first the Cartesian, direct, and strong Cartesian product of hypergraphs are defined. In the next section the definition of the adjacency functions for hypergraphs is introduced and then the theorems on the formation of the adjacency and Laplacian matrices of graph products are applied to hypergraph products. The subsequent section discusses the variable adjacency functions and dynamic graph products. Then some examples are provided from structural mechanics for further clarification of the issues.
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