Approximate eigensolution of Laplacian matrices for locally modified graph products

Laplacian matrices and their spectrum are of great importance in algebraic graph theory. There exist efficient formulations for eigensolutions of the Laplacian matrices associated with a special class of graphs called product graphs. In this paper, the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and/or members, is investigated. The eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model. Implicitly restarted Lanczos method is employed to obtain the required eigenpairs of the reduced problem. Examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the IRL method.