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On Angles and Perturbations of Graphs

On Angles and Perturbations of Graphs PETER ROWLINSON 1. Introduction 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 P +.. . + fi P where to > ... > // : the largest eigenvalue n is called the index of G. x m m m x For ie (1,... , m) andy e {1,...,«} , let cos" (a ) be the angle between the ith eigenspace E(n^ and the yth co-ordinate axis. Thus if e ...,e comprise the standard 15 n orthonormal basis of U then <x = \P e \. The a themselves are customarily referred tj t } w to as angles of G, abusing terminology: their value as graph invariants has been noted 1 1 in [3]. Here we shall consider also the invariants y$, where cos" ^ ) is the angle between P e and P e . Note that the (j, fc)-entry p £ of P is a a y$. t } t k t y Jfc Suppose that G is a spanning subgraph of the graph http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Angles and Perturbations of Graphs

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References (10)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/20.3.193
Publisher site
See Article on Publisher Site

Abstract

PETER ROWLINSON 1. Introduction 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 P +.. . + fi P where to > ... > // : the largest eigenvalue n is called the index of G. x m m m x For ie (1,... , m) andy e {1,...,«} , let cos" (a ) be the angle between the ith eigenspace E(n^ and the yth co-ordinate axis. Thus if e ...,e comprise the standard 15 n orthonormal basis of U then <x = \P e \. The a themselves are customarily referred tj t } w to as angles of G, abusing terminology: their value as graph invariants has been noted 1 1 in [3]. Here we shall consider also the invariants y$, where cos" ^ ) is the angle between P e and P e . Note that the (j, fc)-entry p £ of P is a a y$. t } t k t y Jfc Suppose that G is a spanning subgraph of the graph

Journal

Bulletin of the London Mathematical SocietyWiley

Published: May 1, 1988

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