Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues? Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
 1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\
 1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\
 1 & 1 & 1 & 1 & 1 & 1 & 1 &0\\
\end{array}
\end{pmatrix}\implies\begin{pmatrix}
\begin{array}{ccc}
 0_{2\times2} & 1_{2\times3} & 1_{2\times2}&1_{2\times1}\\
 1_{3\times2} & 0_{3\times3} & 1_{3\times2}&1_{3\times1}  \\
 1_{2\times2} & 1_{2\times3} & 0_{2\times2}&1_{2\times1}  \\
 1_{1\times2} & 1_{1\times3} & 1_{1\times2}&0_{1\times1}  \\
  \end{array}
\end{pmatrix}$ be a symmetric matrix whose entries are blocks(diagonal blocks entry must be zero and non-diagonal blocks entry must be one).  How to prove/disprove that all the eigenvalues of $M$ are not integer if and only if the diagonal blocks are not of the same order.Note that the number of diagonal blocks $\geq 3$ and order of diagonal blocks$\geq 1$ . 

You can have a look at Eigenvalues of complete multipartite graphs. One example with integer spectrum is $K_{2,2,2,8,8\ }.$
