TITLE: Does a symmetric matrix with all entries $0$ or $1$ and with diagonal $0$ have integer eigenvalues?
QUESTION [6 upvotes]: Suppose $A$ is an $n×n$ symmetric matrix with all entries $0$ or $1$, and with diagonal $0$. 
Are all of the eigenvalues of $A$ integers? It works for all the cases I have tried so far.

REPLY [15 votes]: Try $$A=\begin{bmatrix} 0&0&1\\0&0&1\\1&1&0\end{bmatrix}.$$ If I am not wrong its characteristic polynomial is $X(2-X^2)$ and its eigenvalues are $0$, $\sqrt{2}$ and $-\sqrt{2}$.