The Critical Group of the Vertex Corona of Cycles and Complete Graphs in Industry

Xiang Hua Tan; Jian Guo Zhu; Cheng Wen Cai; Kai Rong Yu; Hao Yun Qian

doi:10.4028/www.scientific.net/AMM.357-360.2802

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HomeApplied Mechanics and MaterialsApplied Mechanics and Materials Vols. 357-360The Critical Group of the Vertex Corona of Cycles...

The Critical Group of the Vertex Corona of Cycles and Complete Graphs in Industry

Abstract:

Self-organized criticality is an important theory widely used in various domain such as a variety of industrial accidents, power system, punctuated equilibrium in biology etc.. The Critical Group of the graph is mainly focused on the Abelian sandpile model of self-organized criticality, whose order is the number of spanning trees in the graph, and which is closely connected with the graph Laplacian matrix. In this paper, the main tools will be the computation for Smith normal form of an integer matrix, which can be achieved by the implementation of a series of row and column operations in the ring Ζ of integers. Hence, the structure of the critical group on the vertex corona is determined and it is shown that the Smith normal form is the direct sum of n (m-1)+1cyclic groups. Furthermore, it follows from Kirchooffs Matrix Tree Theorem that the number of spanning trees of the Graph is n (m+1)^{n (m-1)}.

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Periodical:

Applied Mechanics and Materials (Volumes 357-360)

Pages:

2802-2805

DOI:

https://doi.org/10.4028/www.scientific.net/AMM.357-360.2802

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Online since:

August 2013

Authors:

Xiang Hua Tan, Jian Guo Zhu, Cheng Wen Cai, Kai Rong Yu, Hao Yun Qian

Keywords:

Critical Group, Laplacian Matrix, Smith Normal Form, Vertex Corona

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