two previous works, Spectra of Graphs — Theory and Applications [3] and Recent Results in

This volume continues a series of monographs in algebraic graph theory that specialize to spectral graph theory — the study of interconnections between the properties of a graph and the eigen-values and eigenvectors of the associated adja-cency matrix. The common thread between the two previous works, Spectra of Graphs — Theory and Applications [3] and Recent Results in the Theory of Graph Spectra [2], and the volume being reviewed is the presence of Cvetkovic as one of the authors. Spectra of Graphs is a self-contained introduction to the subject that includes references to all of the literature available through 1978. Recent Results in the Theory of Graph Spectra was written as an update, intending to survey all of the literature available through 1984. Eigenspaces of Graphs is different in two respects. While the previous volumes focused on eigenvalues, now, as the title indicates, the focus is on the associated eigenspaces. Second, rather than being an exhaustive survey of previous results the authors are more forward-looking, describing their current and ongoing research into graph angles, star partitions and canonical star bases. The result is a text that is clearly written and instructive (though it is not a textbook) and which should be accessible to experienced researchers and graduate students. In the early days of spectral graph theory it was hoped that knowledge of a graph's eigenval-ues would determine the graph. Since the eigen-values of a graph are independent of the labeling of the vertices it was hoped that the eigenvalues would allow one to easily determine if two graphs were isomorphic. However, the discovery of pairs of non-isomorphic graphs with identical spectra dashed these hopes. As computing power and techniques increased, many more such pairs were found and it became clear that even within restricted classes of graphs the spectra is far from sufficient for characterizing graphs. For example, Schwenk [4] showed that almost every tree has a cospectral mate. Despite being insufficent as a characterization tool, eigenvalues are very helpful as a means of investigating a graph's properties. Translating a question about a graph into a question about its eigenvalues, and then applying techniques from linear algebra, is often a successful approach. Additionally , eigenvalues are useful in a variety of applications, most notably chemistry. It is easy to model a molecule with a graph. (Indeed, some of the early history of graph theory arose in this connection when Cayley …