Abstract
The eigenvalues of a graph are algebraic integers in some algebraic extension of the rationals. We investigate the algebraic degree of these eigenvalues with respect to graph-theoretical properties. We obtain quantitative results showing that a graph with large diameter must have some eigenvalues of large algebraic degree.
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Bang, S., Dubickas, A., Koolen, J.H., Moulton, V.: There are only finitely many distance-regular graphs of fixed valency greater than two. Adv. Math. 269, 1–55 (2015)
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)
Ellenberg, J.S., Venkatesh, A.: The number of extensions of a number field with fixed degree and bounded discriminant. Ann. Math. 163, 723–741 (2006)
Harary, F., Schwenk, A.J.: Which graph have integral spectra? Graphs and Combin. In: Proceedings of Capital Conference, Washington, D.C. 1973. Lecture Notes on Maths. vol. 406, pp. 45–51 (1974)
Hoffman, A.J.: Eigenvalues of graphs, in: “Stud. Graph Theory”, Part II. MAA Stud. Math. 12, 225–245 (1975)
Kronecker, L.: Zwei Sätze über Gleichungen mit ganzzahligen coefficienten. J. Reine Angew. Math. 53, 173–175 (1857)
Mollin, R.A.: Algebraic Number Theory. Chapman and Hall/CRC, Boca Raton (1999)
Robinson, R.M.: Intervals containing infinitely many sets of conjugate algebraic integers. In: Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya, Stanford, pp. 305–315 (1962)
Schmidt, W.M.: Number fields of given degree and bounded discriminant. In: Columbia University Number Theory Seminar, New York, 1992. Paris: Société Mathématique de France, Astérisque 228, 189–195 (1995)
Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. III. Adv. Math. 208, 467–489 (2007)
Tenenbaum, G.: Introduction to analytic and probabilistic number theory. Cambridge University Press, Cambridge (1995)
Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. Springer, New York (1997)
Acknowledgements
The authors are grateful to the anonymous referee for her or his valuable corrections and comments to improve the readability of the article and for introducing us to the Biggs-Smith graph and the doubled Odd graphs.
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Mönius, K., Steuding, J. & Stumpf, P. Which Graphs have Non-integral Spectra?. Graphs and Combinatorics 34, 1507–1518 (2018). https://doi.org/10.1007/s00373-018-1947-z
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DOI: https://doi.org/10.1007/s00373-018-1947-z