Majorisations for the eigenvectors of graph-adjacency matrices
Rahul Dhal, Sandip Roy, Yan Wan, Ali Saberi · International Journal of Control · 2014
We develop majorisation results that characterise changes in eigenvector components of a graph's adjacency matrix when its topology is changed. Specifically, for general (weighted, directed) graphs, we characterise changes in dominant eigenvector components for single- and multi-row incrementations. We also show that topology changes can be tailored to set ratios between the components of the dominant eigenvector. For more limited graph classes (specifically, undirected, and reversibly-structured ones), majorisations for components of the subdominant and other eigenvectors upon graph modifications are also obtained.