Every totally real algebraic integer is a tree eigenvalue

TL;DR

This paper proves that every totally real algebraic integer can be realized as an eigenvalue of a tree graph, providing a new elementary proof and resolving an open problem about the spectrum of random matrices.

Contribution

It offers an independent, elementary proof that every totally real algebraic integer is a tree eigenvalue, strengthening previous results and settling an open problem in spectral graph theory.

Findings

Every totally real algebraic integer is a tree eigenvalue.

The spectrum of certain random matrices has atoms exactly at these algebraic integers.

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).