Neumann minimizers distinct from Dirichlet and boundaryless minimizers

Determine whether there exist weighted or unweighted combinatorial graphs G, together with integers k and parameters p, for which the Neumann spectral minimal energy is attained by a k-partition that is not a minimizer of either the Dirichlet spectral minimal energy or the boundaryless spectral minimal energy.

Background

The paper compares three partition energies on combinatorial graphs: the Dirichlet energy, which imposes zero boundary values; the boundaryless energy, which treats each cluster as an independent induced subgraph; and the Neumann energy, which incorporates Neumann conditions at the cluster boundary. For finite two-ladder graphs and two clusters, the authors show that the Neumann minimal energy agrees with the boundaryless minimal energy, but that the minimizing partition is uniquely the horizontal division into two path graphs, whereas the Dirichlet minimizer has a different configuration.

The unresolved issue asks whether this coincidence and separation behavior is special to the ladder examples, or whether a graph can be constructed where a Neumann-minimizing partition is genuinely different from every minimizer for both of the other two spectral partition problems. The question concerns existence for some graph, number of clusters k, and exponent p.