General characterization of enlarged equitable partitions for graph matrices

Characterize, for large classes of graph matrices M(G), the enlarged equitable partitions (of minimal size when possible) for which the spectrum of the equitable quotient matrix contains all distinct eigenvalues of M(G).

Background

Section 6 introduces Problem \ref{problem distinct 3}, which asks for classes of matrices where an enlarged equitable partition yields a quotient matrix containing all distinct eigenvalues of the parent matrix.

The authors provide constructive examples (e.g., specific unicyclic-like graphs and complete bipartite graphs) where enlarging a cell of an equitable partition restores missing eigenvalues in the quotient, but they state that the problem remains open in general for broad families of graph matrices.