Existence of nearly regular graphs with two Laplacian eigenvalues

Determine, for fixed order, size, and degree-sum-of-squares parameters, how many nearly regular graphs have exactly two distinct nonzero eigenvalues of the Laplacian, and whether any such graphs exist.

Background

The optimization argument in Section 3 shows that a candidate nearly regular graph of maximal complexity should ideally have two distinct nonzero Laplacian eigenvalues. The authors derive formulas for these eigenvalues in terms of their multiplicities, but do not establish which multiplicities are realizable by graphs or whether realizations exist at all for arbitrary parameter choices. This limits the applicability of the spectral optimization method.