Let $G=(V,E)$ is a connected d-regular bipartite graph with the same number of vertices on both sides of the bipartition. It's known that that the largest eigenvalue of its adjacency matrix would be d, and the smallest would be -d.

I was wondering if it was true that if the k-th largest eigenvalue here was $\lambda_k$, then the k-th smallest one would be $-\lambda_k$.

I seem to be able to show that this is true for the second largest eigenvalue (using the fact that the all ones and indicators on each side of the bipartite graph are the largest and smallest eigenvectors, and using that the eigenvalues have multiplicity 1 and then applying the sign flipping trick). However this proof does not seem to hold in general.


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