  Recent work of the authors and their collaborators has uncovered fundamental
connections between the Dirichlet-to-Neumann map, the spectral flow of a
certain family of self-adjoint operators, and the nodal deficiency of a
Laplacian eigenfunction (or an analogous deficiency associated to a
non-bipartite equipartition). Using a refined construction of the
Dirichlet-to-Neumann map, we strengthen all of these results, in particular
getting improved bounds on the nodal deficiency of degenerate eigenfunctions.
Our framework is very general, allowing for non-bipartite partitions,
non-simple eigenvalues, and non-smooth nodal sets. Consequently, the results
can be used in the general study of spectral minimal partitions, not just nodal
partitions of generic Laplacian eigenfunctions.
