It is well known that if $\pi$ is an equitable partition of a graph, then the spectrum of the corresponding partition matrix is a subset of the spectrum of the graph's matrix (where the matrix can be the adjacency, Laplacian, or signless Laplacian - under suitable definitions).

What I'd like to know is whether it is known under which conditions the smallest eigenvalue of the partition matrix is actually the smallest eigenvalue of the graph. One such example that I am aware of is in this paper, where this result is proved for connected threshold graphs: Tam, Bit-Shun; Wu, Shu-Hui On the reduced signless Laplacian spectrum of a degree maximal graph. (English) [J] Linear Algebra Appl. 432, No. 7, 1734-1756 (2010)

Thanks in advance!