Abstract
The spectrum S(G) of a graph G of order p is defined as the non-increasing sequence of the p real eigenvalues of the adjacency matrix of G. It has been found that certain graphs have an integral spectrum, i.e., every eigenvalue is an integer. Thus, it is natural to ask just which graphs have this property. We develop a systematic approach to this question based on operations on graphs. The general problem appears intractable.
Research supported in part by grant 73-2502 from the Air Force Office of Scientific Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Collatz, L., and Sinogowitz, U., Spektren endlicher Graphen, Abh. Math. Sem. Univ. Hamburg, 21 (1957) 64–77.
Harary, F., Graph Theory, Addison-Wesley, Reading, 1969.
Hoffman, A. J., Some Recent Results on Spectral Properties of Graphs, Beitrage zur Graphentheorie (H. Sachs et al., eds.) Teubner, Leipzig (1968), 75–80.
Mowshowitz, A., The Group of a Graph Whose Adjacency Matrix Has All Distinct Eigenvalues, Proof Techniques in Graph Theory, (F. Harary, ed.), Academic Press, New York (1969), 109–111.
Schwenk, A. J., Computing the Characteristic Polynomial of a Graph, this volume p. 153.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin
About this paper
Cite this paper
Harary, F., Schwenk, A.J. (1974). Which graphs have integral spectra?. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066434
Download citation
DOI: https://doi.org/10.1007/BFb0066434
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06854-9
Online ISBN: 978-3-540-37809-9
eBook Packages: Springer Book Archive