A Survey on Spectra of Infinite Graphs

In the last decade, considerable attention has been paid by graph theorists to the study of spectra of graphs and their interaction with structure and characteristic properties of graphs. After the 'classical' book by N. L. Biggs [12] on algebraic graph theory in general, the first comprehensive monograph on this particular topic, by D. M. Cvetkovic, M. Doob and H. Sachs [34], appeared in 1979. It dealt exclusively with finite graphs. In a recent article by the first two authors [32], a survey of new developments is given, including a short section on infinite graphs. On the other hand, spectral theory of graphs, in particular infinite graphs, and related topics have been dealt with to a considerable extent 'disguised' in the framework of the theory of nonnegative matrices, harmonic analysis on graphs and discrete groups (Cayley graphs), analytic probability theory, Markov chains and other mathematical branches. In fact, some results have been rediscovered several times under different viewpoints. The purpose of this survey is to give an overview of results on spectra of infinite graphs, emphasizing how contributions from different areas fit into this graphtheoretical setting. For the moment, we point out the books by E. Seneta [113] and by A. Figa-Talamanca and M. A. Picardello [45] as two examples. Among the variety of books on the background in functional analysis and matrix theory, we emphasize [1, 30, 41, 118, 129]. Our approach follows that of B. Mohar [89] and related definitions of spectra, as they have been used in the mathematical fields mentioned above. A different approach, due to A. Torgasev [119], will be only briefly discussed; the reader is referred to the recent book [33]. Primarily, this survey is addressed to graph theorists; there also is some emphasis on ' spectral' properties of random walks on graphs. The selection of those topics which can (or should) in some way be regarded in connection with spectral theory of graphs could most likely be extended towards infinity. Therefore, the present survey is biased by the viewpoint of the authors and cannot be complete. We apologize to all those who feel that their work is missing in the references or has not been emphasized sufficiently in the text.