Abstract
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.
Highlights
There are different possibilities and as we will see, diverse, arguably natural regularity classes for metric graph partitions
The interaction of these eigenvalues—especially those other than Dirichlet—with the vertices can become delicate. For this reason, in past studies of nodal domains one would usually assume that the eigenfunctions do not vanish at the vertices—a condition which is generically satisfied but cannot be assumed when studying optimisation problems, since this condition is not stable under taking limits. This condition was assumed in possibly the only previous work on spectral partitions of metric graphs to date [4], which focused on the rather different question of optimality properties of partitions induced by such nodal domains under this assumption
After introducing our various natural classes of partitions, will be to prove existence of partitions of minimal energy, upon minimising in several such regularity classes; and to observe that certain, new, classes arise most naturally, as the classes which always contain their minima. All this requires a whole new theory, which we illustrate in Sect. 2: after a brief reminder on metric graphs and associated function spaces and Laplacians, we introduce the aforementioned classes of partitions, the two most important of which we call connected and rigid
Summary
Page 3 of 63 61 subsets, here subgraphs, which we will always call clusters. There are different possibilities and as we will see, diverse, arguably natural regularity classes for metric graph partitions. Such a connection suggests a second source of inspiration for us here, namely studies of nodal domains for discrete graph Laplacians: we mention in particular [18], whose results have been recently extended to the case of general quadratic forms generating positive semigroups in [27] In yet another direction, related to Cheeger partitions and free discontinuity problems, the authors of [14] proved existence and some regularity for minimal partitions associated with the Robin Laplacian. The interaction of these eigenvalues—especially those other than Dirichlet—with the vertices can become delicate For this reason, in past studies of nodal domains one would usually assume that the eigenfunctions do not vanish at the vertices—a condition which is generically satisfied but cannot be assumed when studying optimisation problems, since this condition is not stable under taking limits (cf [5]). This condition was assumed in possibly the only previous work on spectral partitions of metric graphs to date [4], which focused on the rather different question of optimality properties of partitions induced by such nodal domains under this assumption
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