Graph–Group Method for the Analysis of Symmetric-Regular Structures

Abstract

In this chapter, a combined graph–group method is presented for eigensolution of special graphs. The study of symmetric graphs with regularity is the main objective of this study. Many structural models are regular and usually have symmetric configurations. Here, the proposed method operates symmetry analysis of the entire structure utilising the symmetry properties of its simple generators. The model of a structure is considered as a product graph, and the Laplacian matrix, as one of the most important matrices associated with a graph, is studied. Characteristic problem of this matrix is investigated using symmetry analysis via group theory enriched by graph theory. The decomposition of Laplacian matrix of such graphs is performed in a step-by-step manner, based on the presented method. This method focuses on simple paths which generate large networks and finds the eigenvalues of the network using the analysis of the simple generators. Group theory is utilised as the main tool, improved by some concepts of graph products. As an application of the method, a benchmark problem of group theory from structural mechanics is studied. Vibration of cable nets is analysed and the frequencies of the networks are calculated using a hybrid graph–group method [1].