Suppose that $G$ is a connected graph with equitable partition $\pi$. Then the eigenvalues of the *divisor multigraph* $G / \pi$ are all eigenvalues of $G$. (Perhaps excluding some pathological cases) the largest eigenvalue of $G/\pi$ is the Perron value of $G$ and thus simple in the spectrum of $G$. 

I would like to know if it's always true that the smallest eigenvalue of $G/\pi$ is also simple in the spectrum of $G$. 


  [1]: http://mathoverflow.net/questions/96275/equitable-partitions