Eigensolution of augmented graph products using shifted inverse iteration method

Abstract

SUMMARYTwo important matrices associated with graphs are adjacency and Laplacian matrices. In this paper efficient methods are presented for eigensolution of graph products augmented by other graphs. For augmentations that do not destroy the symmetry of the graph products, a method is proposed for decomposition of matrices resulting in considerable simplification of their eigensolution. For graphs composed of two non-overlapping graph products joined through a small number of link members, a method based on shifted inverse iteration is proposed which utilizes all eigenvalues and eigenvectors of each individual graph products. Owing to the availability of fast methods for eigensolution of graph products, this method simplifies the eigensolution of a variety of graph models and proves to be very efficient in determining the few smallest eigenpairs of these models with high levels of accuracy.

Canonical Forms for Symmetric and Regular Structures

“…The general idea is that such forms may not be explicitly decomposed into n decoupled diagonal blocks; however efficient methods may be devised in order to relate these forms or the corresponding mechanical systems to more convenient forms or systems. The authors have an ongoing research on this area and have employed different techniques to exploit the potential of matrix canonical forms in the analysis of nonconforming cases similar to this one [58][59][60]. In Section 4, a method is presented for efficient handling of regular structures exhibiting matrix patterns similar to Eq.…”