每个全实代数整数是树eigenvalue
Every totally real algebraic integer is a tree eigenvalue
摘要：
图特征值是全实代数整数的例子，即具有整数系数的实根首一多项式的根。相反，每个全实代数整数都作为某个有限图的特征值出现的事实是一个深刻的结果，40年前由霍夫曼提出，17年后由埃斯蒂斯证明。这个简短的文件提供了一个独立的和基本的证明，一个更强的声明，即图实际上可以选择是一棵树。作为一个副产品，我们的结果意味着具有独立Bernoulli $\,\left(\frac{c}{n}\right)$元的$n\times n$对称矩阵（$c>0$固定为$n\to\infty$）的极限谱的原子恰好是全实代数整数。这解决了Ben Arous（2010）提出的一个开放性问题。
英文摘要：
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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