Existence and enumeration of two-eigenvalue nearly regular graphs

Determine how many nearly regular graphs with fixed order, size, and two distinct nonzero Laplacian eigenvalues exist, including whether any such graphs exist for the specified spectral parameters.

Background

The paper uses Lagrange-multiplier arguments to show that an ideal maximizer within the nearly regular class should have exactly two distinct nonzero Laplacian eigenvalues. The two eigenvalues and their multiplicities are constrained by the order, degree data, and the first two Laplacian trace values.

Although several examples are identified, the authors do not classify or even determine the existence and number of graphs realizing the resulting eigenstructure for arbitrary fixed parameters. This is a concrete unresolved spectral-realizability question relevant to identifying all maximal-complexity graphs.