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Article Dans Une Revue Journal of Combinatorial Theory, Series B Année : 2015

Every totally real algebraic integer is a tree eigenvalue

Résumé

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).
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Dates et versions

hal-00789806 , version 1 (18-02-2013)
hal-00789806 , version 2 (04-09-2014)

Identifiants

Citer

Justin Salez. Every totally real algebraic integer is a tree eigenvalue. Journal of Combinatorial Theory, Series B, 2015, 111, pp.249-256. ⟨10.1016/j.jctb.2014.09.001⟩. ⟨hal-00789806v2⟩
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