  We study partition problems based on two ostensibly different kinds of energy
functionals defined on $k$-partitions of metric graphs: Cheeger-type
functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the
corresponding values are the $k$-Cheeger constants of the graph; and
functionals built using the first eigenvalue of the Laplacian with positive,
i.e. absorbing, Robin (delta) vertex conditions at the boundary of the
partition elements. We prove existence of minimising $k$-partitions, $k \geq
2$, for both these functionals. We also show that, for each $k \geq 2$, as the
Robin parameter $\alpha \to 0$, up to a renormalisation the spectral minimal
Robin energy converges to the $k$-Cheeger constant. Moreover, up to a
subsequence, the Robin spectral minimal $k$-partitions converge in a natural
sense to a $k$-Cheeger cut of the graph. Finally, we show that as $\alpha \to
\infty$ there is convergence in a similar sense to the corresponding Dirichlet
minimal energy and partitions.
  It is strongly expected that similar results hold on general (smooth,
bounded) Euclidean domains and manifolds.

Spectral Theory