Current research on the spectral theory of finite graphs may be
seen as part of a wider effort to forge closer links between
algebra and combinatorics (in particular between linear algebra and
graph theory).This book describes how this topic can be
strengthened by exploiting properties of the eigenspaces of
adjacency matrices associated with a graph. The extension of
spectral techniques proceeds at three levels: using eigenvectors
associated with an arbitrary labelling of graph vertices, using
geometrical invariants of eigenspaces such as graph angles and main
angles, and introducing certain kinds of canonical eigenvectors by
means of star partitions and star bases. One objective is to
describe graphs by algebraic means as far as possible, and the book
discusses the Ulam reconstruction conjecture and the graph
isomorphism problem in this context. Further problems of graph
reconstruction and identification are used to illustrate the
importance of graph angles and star partitions in relation to graph
structure. Specialists in graph theory will welcome this treatment
of important new research.
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