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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?

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up vote 4 down vote accepted

There are infinitely many counterexamples. Let $\pi$ be the distance partition relative to a vertex in a strongly regular graph. Then $\pi$ is equitable and $G/\pi$ is a path, so its eigenvalues are all simple. But if $G$ is not bipartite, its least eigenvalue is not simple.

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