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).