On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs

We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of nodal points and explore conditions under which Neumann domains of eigenfunctions correspond to minimizers of a class of spectral partition problems, often known as spectral minimal partitions. Our main focus is the analysis on tree graphs, where we characterize the spectral energies of such partitions and relate them to the eigenvalues of the Laplacian under genericity assumptions. Notably, we introduce a notion analogous to Courant-sharpness for Neumann counts and identify when spectral minimal partitions coincide with partitions formed by Neumann domains of eigenfunctions.