Majorisations for the eigenvectors of graph-adjacency matrices

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 singleand 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 reversiblystructured ones), majorisations for components of the subdominant and other eigenvectors upon graph modifications are also obtained.