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Graph eigenvalues are examples of totally real algebraic integers, i.e. rootsof real-rooted monic polynomials with integer coefficients. Conversely, thefact that every totally real algebraic integer occurs as an eigenvalue of somefinite graph is a deep result, conjectured forty years ago by Hoffman, andproved seventeen years later by Estes. This short paper provides an independentand elementary proof of a stronger statement, namely that the graph mayactually be chosen to be a tree. As a by-product, our result implies that theatoms of the limiting spectrum of $n\times n$ symmetric matrices withindependent Bernoulli$\,\left(\frac{c}{n}\right)$ entries ($c>0$ is fixed as$n\to\infty$) are exactly the totally real algebraic integers. This settles anopen problem raised by Ben Arous (2010).

Every totally real algebraic integer is a tree eigenvalue