On Angles and Perturbations of Graphs

We consider only finite undirected graphs without loops or multiple edges. Let G be a non-trivial connected graph whose vertices are labelled 1,2,..., n and let A be the corresponding (0, l)-adjacency matrix of G. Let A have spectral form to Px + . . . + fim Pm where to > ... > //m: the largest eigenvalue nx is called the index of G. For ie ( 1 , . . . , m) andye {1,...,«}, let cos" (ay) be the angle between the ith eigenspace E(n^ and the yth co-ordinate axis. Thus if e15 . . . ,e n comprise the standard orthonormal basis of U then 1 such that for complex £ with |£| < R, there exist analytic functions ^ ( 0 * (O such that