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

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).