Abstract
There are different matrices associated with a graph, such as incidence matrix, the adjacency matrix and the Laplacian matrix. One of the aims of algebraic graph theory is to determine how properties of graphs are reflected in algebraic properties of these matrices. The eigenvalues and eigenvectors of these matrices provide valuable tools for combinatorial optimisation and in particular for ordering of sparse symmetric matrices such as the stiffness and flexibility matrices of the structures. Here, algebraic graph-theoretical methods and metaheuristic-based algorithms are provided for nodal ordering for bandwidth and profile reduction.
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References
Kaveh A, Sharafi P (2008) Optimal priority functions for profile reduction using ant colony optimization. Finite Elem Anal Des 44(3):131–143
Kaveh A, Sharafi P (2008) Nodal ordering for bandwidth reduction using ant system algorithm. Eng Comput 26(3):313–337
Kaveh A, Sharafi P (2012) Ordering for bandwidth and profile minimization problems via charged system search method. Iran J Sci Technol 36:39–52
Gould P (1967) The geographical interpretation of eigenvalues. Trans Inst Br Geogr 42:53–58
Kaveh A (2004) Structural mechanics: graph and matrix methods, 3rd edn. Research Studies Press, Somerset
Grimes RG, Pierce DJ, Simon HD (1990) A new algorithm for finding a pseudo-peripheral node in a graph. SIAM J Anal Appl 11:323–334
Schwenk AJ, Wilson RJ (1978) On the eigenvalues of a graph. In: Beineke LW, Wilson RJ (eds) Selected topics in graph theory. Academic Press, New York
Straffing PD (1980) Linear algebra in geography; eigenvectors of networks. Math Mag 53:269–276
Hall K (1970) R-dimensional quadratic placement algorithm. Manag Sci 17:219–229
Fiedler M (1973) Algebraic connectivity of graphs. Czech Math J 23:298–305
Paulino GH, Menezes IFM, Gattass M, Mukherjee S (1994) Node and element resequencing using the Laplacian of a finite element graph: part I—general concepts and algorithms. Int J Numer Methods Eng 37:1511–1530
Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search. Acta Mech 213(3–4):267–286
Halliday D, Resnick R, Walker J (2008) Fundamentals of physics, 8th edn. Wiley, Hoboken
Sloan SW (1986) An algorithm for profile and wavefront reduction of sparse matrices. Int J Numer Methods Eng 23:1693–1704
King IP (1970) An automatic reordering scheme for simultaneous equations derived from network systems. Int J Numer Methods Eng 2:523–533
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Kaveh, A. (2014). Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory and Meta-heuristic Based Methods. In: Computational Structural Analysis and Finite Element Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-02964-1_5
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DOI: https://doi.org/10.1007/978-3-319-02964-1_5
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