Every totally real algebraic integer is a tree eigenvalue

Abstract : Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and elementary proof of a stronger statement, namely that the graph may actually be chosen to be a tree. As a by-product, our result implies that the atoms of the limiting spectrum of $n\times n$ symmetric matrices with independent Bernoulli$\,\left(\frac{c}{n}\right)$ entries ($c>0$ is fixed as $n\to\infty$) are exactly the totally real algebraic integers. This settles an open problem raised by Ben Arous (2010).
Type de document :
Pré-publication, Document de travail
2013
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https://hal.archives-ouvertes.fr/hal-00789806
Contributeur : Justin Salez <>
Soumis le : jeudi 4 septembre 2014 - 14:40:53
Dernière modification le : jeudi 27 avril 2017 - 09:46:19
Document(s) archivé(s) le : vendredi 5 décembre 2014 - 10:28:50

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  • HAL Id : hal-00789806, version 2
  • ARXIV : 1302.4423

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UPMC | INSMI | USPC | PMA

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Justin Salez. Every totally real algebraic integer is a tree eigenvalue. 2013. <hal-00789806v2>

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