Abstract
In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest such semigroup seems to be of independent interest.
Similar content being viewed by others
References
Bass, H., Estes, D.R., Guralnick, R.M.: Eigenvalues of symmetric matrices and graphs. J. Algebra 168, 536–567 (1994)
Biggs, N.: Algebraic Graph Theory, Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (1993)
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs Universitext. Springer, New York (2012)
Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs, 3rd edn. Johann Ambrosius Barth, Leipzig (1995)
Estes, D.R.: Eigenvalues of symmetric integer matrices. J. Number Theory 42, 292–296 (1992)
Hoffman, A.J.: Eigenvalues of graphs. In: Studies in graph theory, Part II, Studies in Math., Vol. 12, Mathematical Association of America, Washington, DC, pp. 225–245 (1975)
Knuth, D.E.: The Art of Computer Programming, Vol. 1: Fundamental Algorithms. Addison-Wesley, Reading (1997)
Motzkin, T.: From among \(n\) conjugate algebraic integers, \(n-1\) can be approximately given. Bull. Am. Math. Soc. 53, 156–162 (1947)
Renteln, P.: On the spectrum of the perfect matching derangement graph. J. Algebraic Combin. 56, 215–228 (2022)
Rowlinson, P.: The main eigenvalues of a graph: a survey. Appl. Anal. Discrete Math. 1, 445–471 (2007)
Salez, J.: Every totally real algebraic integer is a tree eigenvalue. J. Combin. Theory Ser. B 111, 249–256 (2015)
Tian, F., Wang, Y.: On the multiplicity of positive eigenvalues of a graph. Linear Algebra Appl. 652, 105–124 (2022)
Yu, Q., Liu, F., Zhang, H., Heng, Z.: Note on graphs with irreducible characteristic polynomials. Linear Algebra Appl. 629, 72–86 (2021)
Zhang, Y., Zhou, Q., Wong, D.: A note on the multiplicities of the eigenvalues of a tree. Linear Algebra Appl. 652, 97–104 (2022)
Acknowledgements
I thank the referees for pointing out several inaccuracies.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dubickas, A. An Upper Bound for the Height of a Tree with a Given Eigenvalue. Combinatorica 44, 299–310 (2024). https://doi.org/10.1007/s00493-023-00071-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-023-00071-2