Improved group theoretic method using graph products for the analysis of symmetric-regular structures

In this paper, a new combined graph-group method is proposed for eigensolution of special graphs. Symmetric regular graphs are the subject of this study. Many structural models can be viewed as the product of two or three simple graphs. Such models are called regular, and usually have symmetric configurations. The proposed method of this paper performs the symmetry analysis of the entire structure via symmetric properties of its simple generators. Here, a graph is considered as the general model of an arbitrary structure. The Laplacian matrix, as one of the most important matrices associated with a graph, is studied in this paper. The characteristic problem of this matrix is investigated using symmetry analysis via group theory enriched by graph theory. The method is developed and decomposition of the Laplacian matrix of such graphs is studied in a step-by-step manner, based on the proposed method. This method focuses on simple paths which generate large networks, and finds the eigenvalues of the network via analysis of the simple generators. Group theory is the main tool, which is improved using the concept of graph products. As a mechanical application of the method, a benchmark problem of group theory in structural mechanics is studied in this paper. Vibration of cable nets is analyzed and the frequencies of the networks are calculated using the combined graph-group method.