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 $G$.
EDIT: Turns out, I have a counterexample to this. Should I close the question?