A theory of spectral partitions of metric graphs
James B. Kennedy, Pavel Kurasov, Corentin Léna, Delio Mugnolo
aa r X i v : . [ m a t h . SP ] M a y A THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS
JAMES B. KENNEDY, PAVEL KURASOV, CORENTIN L´ENA, AND DELIO MUGNOLO
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 clustersthus arising and use their spectral gaps to define several notions of partition energies; thisis the graph counterpart of the well-known theory of spectral minimal partitions on planardomains and includes the setting in [Band et al , Comm. Math. Phys. (2012), 815–838] asa special case. We focus on the existence of optimisers for a large class of functionals definedon 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 nodalpartitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allowsfor explicit computation and analytic – rather than numerical – results. Not only do we recoverthe main assertions in the theory of spectral minimal partitions on domains, as studied in[Conti et al , Calc. Var. (2005), 45–72; Helffer et al , Ann. Inst. Henri Poincar´e Anal. NonLin´eaire (2009), 101–138], but we can also generalise some of them and answer (the graphcounterparts of) a few open questions. Contents
1. Introduction 22. Graphs and partitions 52.1. Basic definitions 52.2. A motivating example 92.3. Graph partitions 113. Topological issues of graph partitions 163.1. Primitive partitions 163.2. Partition convergence 173.3. Existence results for energy functionals 204. Existence of spectral minimal partitions 215. Nodal and bipartite minimal Dirichlet partitions 28
Mathematics Subject Classification.
Key words and phrases.
Quantum graphs, Laplace operators, eigenvalues, eigenfunctions, spectral minimalpartitions.The authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld withinthe framework of the cooperation group on “Discrete and continuous models in the theory of networks”. All theauthors would like to acknowledge networking support by the COST Action CA18232. The work of J.B.K. wassupported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal, via the program “Investigador FCT”, ref-erence IF/01461/2015, and via project PTDC/MAT-CAL/4334/2014. P.K. was also supported by the SwedishResearch Council grant D0497301. C.L. was partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia,Portugal (under the project OPTFORMA, IF/00177/2013) and the Swedish Research Council (under the grantD0497301). D.M. was also supported by the Deutsche Forschungsgemeinschaft (Grant 397230547). The authorswould like to thank Matthias Hofmann (Lisbon) and Marvin Pl¨umer (Hagen) for a number of helpful commentsand suggestions, including M.H. for the idea for Example 8.8. p L Nk,p , L Dk,p , M Nk and M Dk Introduction
How to cut a connected object – be it a manifold, a domain, or a graph – into k mutuallydisjoint, connected parts? Partitioning of objects is a natural topic in geometry and has impor-tant consequences in applied sciences, like data analysis or image segmentation. This article isdevoted to the issue of graph partitioning via properties of eigenvalues.Consider a connected, compact metric graph [BK13, Mug14, Kur20]: the lowest eigenvalueof the Laplacian with natural vertex conditions is 0, with the constant functions as associatedeigenfunctions. Thus, all further eigenfunctions are orthogonal to the constants and hencesign-changing: it is conceivable to use the support of their positive and negative parts as anatural splitting of the graph, similar to the classical Cheeger approach, and indeed this hasbeen discussed by several authors [Nic87, Pos12, Kur13, DPR16, KM16].More generally, one can consider the eigenfunctions associated with the k -th eigenvalue and,inspired by Sturm’s Oscillation Theorem (or in higher dimensions by Courant’s Nodal DomainTheorem), hope that they deliver a reasonable splitting into k subsets, which we then interpretas clusters . However, a priori there is no reason for this splitting to result in k pieces: thezeros of the eigenfunction may divide the graph into fewer – or more – than k pieces, or nodaldomains. (Similar issues have been observed in the case of discrete graphs by Davies et al.in [DGLS01], and recently extended to the case of general quadratic forms generating positivesemigroups in [KS19].) Indeed, accurate estimates on the number ν k of nodal domains of a k -theigenfunction involve the topology of the metric graph (via its first Betti number) and havebeen proved in [GSW04, Ber08, Ban14], see [BK13, § domain Ω ⊂ R into precisely k connectedpieces was introduced in [CTV05] and amounts to seeking those k -partitions (Ω i ) ≤ i ≤ k of Ωsuch that, upon imposing Dirichlet conditions at the boundary of each Ω i and considering thelowest eigenvalue λ (Ω i ) of the Dirichlet Laplacian on Ω i , a certain function – typically themaximum – of the vector ( λ (Ω i )) ≤ i ≤ k is minimised. The attention devoted to such spectralminimal partitions was greatly boosted when a connection to nodal domains `a la Courant was THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 3 established in [HHOT09]. In another direction, related to Cheeger partitions and free discon-tinuity problems, the authors of [BFG18] proved existence and some regularity for minimalpartitions associated with the Robin Laplacian.The aim of the present article is to develop a comprehensive theory of spectral partitions onmetric graphs. In the case of planar domains, whenever defining a spectral partition the onlyissue is to make sure that each subdomain can be associated with a Dirichlet eigenvalue; hence,it is sufficient and also natural to consider those partitionings consisting of open, mutuallydisjoint sets Ω i whose closures are contained in the closure of Ω; the boundaries of nodaldomains turn out to be smooth curves – along which Dirichlet conditions are imposed – meetingat a finite number of points. But when it comes to metric graphs, there is no such naturaldefinition of partitioning. A first major issue is how to proceed when cutting through vertices,more specifically: what transformations of the connectivity of a piece, or cluster , G i shouldbe allowed for, should G i contain a vertex that (as a vertex in the given metric graph G ) liesat the boundary between it and a further cluster G j . While the few existing works that haveconsidered metric graph partitioning, in particular [BBRS12], avoided this problem by onlyallowing for cuts through the interiors of edges, or equivalently vertices of degree 2, we willdefine several “partitioning” rules of increasing generality, based on which types of cuts maybe allowed: the most important ones give rise to what we call rigid and loose partitions. Thisrequires a whole new theory, which we illustrate in Section 2.3 with the help of a few elementaryexamples the most ubiquitous of which is a lasso graph, demonstrating that loose and rigidpartitions may be considered natural objects.The main existence result in the theory of spectral minimal partitions on domains was ob-tained in [CTV05, §
2] by variational methods; a similar – but technically much more involved– approach was recently used in [BFG18] to deal with the eigenvalues of Robin Laplacians.The natural constraints studied there are however too rough for our setting, and we have toproceed differently. Given a compact metric graph G , we introduce in Section 3 a Polish metricspace P ( G ) of its partitions, study general lower semi-continuous functionals with respect to theinduced topology, and discuss qualitative properties of their minima. Our abstract approachpays off: among other things, loose and rigid partitions are indeed natural simply because –unlike many other, ostensibly natural, classes – they define closed subsets of the metric space P ( G ) and are hence particularly suitable for minimisation purposes.This theoretical toolbox is then applied in Section 4 to several classes of optimal partitionproblems, including those appearing in earlier studies on nodal partitions, like [BBRS12]. Bychecking that the relevant energy functionals actually satisfy the (rather mild) sufficient condi-tions introduced in Section 3.3, we can finally prove existence of optimal partitions for Lapla-cians with either Dirichlet or natural vertex conditions. (Needless to say, minimising amongrigid or loose partitions will generally yield different optima, an issue we will touch upon inSection 8.1.) Our investigations show that both optimisation probelms are well-motivated.Dirichlet conditions at the cut points – the classical choice in the earlier literature, both ondomains and metric graphs – are naturally related to the issue of nodal domains, a connectionthat led to the very birth of this field in [CTV05]. Imposing natural conditions at the cutpoints, on the other hand, will be shown to lead to well-posed spectral problems whose min-imizing partitions consist of clusters that are connected in a more straightforward, apparentsense. These results can be further generalised considering graph counterparts of the energyfunctionals first introduced in [CTV05] – essentially, the mean value of p -th powers of spectral J. B. KENNEDY, P. KURASOV, C. L´ENA, AND D. MUGNOLO gaps of a suitable Laplacian, defined clusterwise; this amounts to studying minima of func-tionals Λ Dp and Λ Np , p ∈ (0 , ∞ ], defined on the partition space P ( G ). Similar ideas may alsobe developed if more general conditions – say, δ -couplings – should imposed at the cut points,analogously to what was done in [BFG18] for the case of domains, although we will not de-velop such ideas here. Neumann domains, a Neumann-type analogue of nodal domains, havebeen studied recently on quantum graphs [AB19, ABBE18], and it is natural to ask whetherthere is a similar link between these and Neumann-type partitions as there is between Dirichletpartitions and nodal domains. We also leave this question to future work. Also, one could inprinciple study other spectral quantities, for example by considering higher eigenvalues. Weleave these as open problems to be discussed in later investigations.Beginning with [HHOT09], much research has been devoted not just to the issue of regularityof spectral minimal partitions of planar domains, but also to the shape of the partition elements.In view of the Faber–Krahn inequality, all subdomains of Ω would try to get as close as possibleto a disc in order to minimise the lowest Dirichlet eigenvalue. A spectral minimal k -partitionis such that all these subdomains can, roughly speaking, find an optimal compromise: this isconjectured to lead asymptotically (for large k ) to a hexagonal tiling of Ω, see [BNH17, § graph partitions with respect to the energy functionals Λ Dp and Λ Np look like?Section 5 is devoted to this question. Some properties of spectral minimal partitions for theDirichlet Laplacian have been already discussed by Band et al [BBRS12] under the assumptionthat they actually exist , and provided there is a ground state that does not vanish at anyvertex. We can generalise some of their findings by dropping this structural assumption, whichin fact typically fails if the graph is highly symmetric. While one simple optimisation problemdoes actually deliver a 2-partition that agrees precisely with the nodal domains of a secondeigenfunction of the Laplacian with natural transmission conditions, an exact characterisationof k -partitions minimising any reasonable energy functional for k ≥ k large enough, any optimalpartition will consist of a collection of stars (which in view of their minimising properties [Fri05]can be regarded as graph analogues of disks, or hexagons). We content ourselves with showingthat the interplay between Dirichlet minimal partitions and nodal domains does hold for trees,at least; we briefly summarise similarities and differences between the qualitative properties ofspectral minimal partitions on planar domains and metric graphs in Section 5.6.To show the flexibility of our approach, analogous spectral partitions that maximise twodifferent energy functionals Ξ Dp and Ξ Np are discussed in Section 6 by showing that they alsosatisfy our basic topological assumptions.In Section 7 we turn to the issue of the dependence of minimisers of Λ Dp and Λ Np on p , andalso on the edge lengths of the underlying metric graph G , for a fixed topology. Here thesimplicity of the 1-dimensional setting of metric graphs is highly advantageous: while such a p -dependence is only numerically observed in the case of domains, we study in detail spectralminimal partitions of a 3-star and show their dependence of p in an analytic way.Finally, in Section 8 we present examples comparing the different optimisation problems (forΛ Dp , Λ Np , Ξ Dp and Ξ Np ) and the corresponding optimal energies on a fixed graph: these differentproblems tend, naturally, to split the graph in different ways. Here we present a couple ofheuristic conjectures based on our examples; in future work we intend to return to the questionof how these different problems behave in a more rigorous and complete way. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 5
To enhance the readability of the article and for ease of reference, the following table collectsa number of new notions and symbols used throughout the paper.Symbol Description/name See G , e G , G ′ metric graph, ur-graph Sec. 2.1, Def. 2.3canonical representative Def. 2.3 G underlying discrete (ur-)graph Def. 2.1, Def. 2.3 λ ( G ), λ ( G ; V D ) first Dirichlet eigenvalue Eq. (2.4) µ ( G ) first nontrivial natural eigenvalue Eq. (2.5)cut of a graph Def. 2.6(nontrivial) cut through a vertex Def. 2.7 P = {G , . . . , G k } ( k -)partition Def. 2.8 G i cluster of a partition Def. 2.8Ω i cluster support corresponding to G i Def. 2.9Ω partition support Def. 2.9 C ( P ) cut set (cut points) of P Def. 2.10 V D ( G i ) set of cut points in G i Def. 2.10 ∂ P separation set (points) of P Def. 2.10neighbour, neighbouring cluster Def. 2.11loose, rigid, faithful, internallyconnected, proper partition Def. 2.12 P , P k set of loose ( k -)partitions Eq. (2.10) R , R k set of rigid ( k -)partitions Eq. (2.10) ρ Ω i set of possible rigid clusters for Ω i Eq. (2.11) C cut pattern, similar partition Def. 3.1 T , T C primitive partition (associated with cut pattern C ) Def. 3.3Γ G set of ur-graphs for G Sec. 3.2 d Γ G ( G , e G ) distance between metric graphswith same discrete ur-graph Sec. 3.2[ v ] equivalence class of vertices converging to v Sec. 3.2(strongly) lower semi-continuous Def. 3.12Λ Dp ( P ), Λ Np ( P ) Dirichlet, natural partition energy Eq. (4.1), Eq. (4.2) L Dk,p ( G ), L Nk,p ( G ) rigid Dirichlet, natural minimal energy Eq. (4.3) e L Dk,p ( G ), e L Nk,p ( G ) loose Dirichlet, natural minimal energy Eq. (4.3)Dirichlet, natural ( k -)equipartition Def. 5.1nodal, generalised nodal, bipartite partition Def. 5.2, Def. 5.3, Def. 5.8 ν , ν ( ψ ) number of nodal domains (of ψ ) Prop. 5.7Ξ D ( P ), Ξ N ( P ) min. Dirichlet, natural partition energy Eq. (6.2), Eq. (6.1) M Dk ( G ), M Nk ( G ) Dirichlet, natural max-min energy Eq. (6.3)2. Graphs and partitions
Basic definitions.
We start with the metric graphs we shall be considering; it will benecessary to consider the formalism we will be using in some detail, which mirrors the oneused in [KS02]. By a metric graph G = ( V , E ) we understand a pair consisting of a vertex set J. B. KENNEDY, P. KURASOV, C. L´ENA, AND D. MUGNOLO V = V ( G ) = { v , . . . , v N } and an edge set E = E ( G ) = { e , . . . , e M } ; throughout the paper wewill always assume these to be finite sets.Each edge e m = e m ( G ), m = 1 , . . . , M , is identified with a compact interval [ x m − , x m ] ⊂ R of length | e m | = x m − x m − belonging to a separate copy of R . Each edge should connect twovertices: formally, we introduce an equivalence relation on the set of endpoints { x j } Mj =1 , thuspartitioning it into nonempty, mutually disjoint sets(2.1) { x j } Mj =1 = V ∪ . . . ∪ V N . The vertex v n ∈ V ( G ) is identified with the set V n = V n ( G ).If x m − , x m ∈ V m ∪ V m for some m , then we write e m ≡ v m v m and in this case we saythat e m is incident with the vertices v m , v m . We refer to the cardinality of V n as the degree of v n , written deg v n .Vertices of degree two are allowed, but are called dummy vertices ; these can be introducedand removed at will without altering any of the properties of the graph (in particular thespectral quantities) in which we will be interested, as we shall discuss below. Loops , that is,edges incident with only one vertex, and multiple edges, that is, distinct edges incident withthe same pair of vertices, are also allowed.Any metric graph G = ( V , E ) will be identified with a set of equivalence classes of points byextending the equivalence relation (2.1) to all points in the interior of each edge (interval); thiswill be done by associating with any x ∈ int e m = ( x m − , x m ) equivalence class { x } formed byone element. With this in mind, in future we will take points x ∈ G , and in particular regardthe vertices v n as points in the set, without further comment. However , for some purposes it isimportant to remember that v n and V n are different objects; indeed, our theory relies essentiallyupon the possibility to cut through a vertex v n by subdividing V n into two or more nonempty,mutually disjoint subsets.We will write |G| = P e ∈E | e | = P Mm =1 | e m | for the finite total length of the graph, the sum ofthe lengths of the edges. We refer to the monographs [BK13, Mug14] for more information onmetric graphs in general.A metric graph has both an underlying discrete structure and a notion of distance definedon it, and both will be important to us. The interested reader may take [Die05] as a stan-dard reference for classical notions and results in combinatorial graph theory (e.g., regarding subdivisions as will appear). Definition 2.1.
Given a metric graph G = ( V , E ) the underlying discrete graph (or associateddiscrete graph ) is the discrete graph G = ( V , E ) for which there are bijections Φ : V → V andΨ : E → E such that for all e ∈ E and all e ∈ E , Ψ( e ) = e implies the vertices incident with e in G are mapped by Φ to the vertices incident with e in G . If Ψ( e ) = e , then we say that e and e correspond to each other.We next recall how a canonical distance is introduced on metric graphs. Given x, y ∈ G ,we take dist( x, y ) to be the minimal length among all paths connecting x with y , see [Mug14,Def. 3.14] for details. If the graph is not connected then we set the distance between pointsbelonging to different connected components to be infinity. Throughout this paper we alwaysconsider on G the topology induced by this distance: in particular, given a subset Ω of G wecan consider its interiorint Ω := { x ∈ Ω : { y ∈ G : dist( x, y ) < ε } ⊂ Ω for some ε > } , THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 7 and its boundary(2.2) ∂ Ω := Ω \ int Ω . (Regarding terminology, we consider ∂ Ω to consist of those points that separate
Ω from
G \
Ω.)Equipped with the distance function, each metric graph is a metric space.Summarising, we shall assume throughout that:
Assumption 2.2.
The metric graph G is finite compact and connected, i.e. the vertex set isfinite, the edge set is finite, each edge has finite length and there is a continuous path connectingany two points on the graph.The only assumption we will occasionally have cause to drop is of connectedness, but in suchcases we will always state this explicitly. Isometric isomorphisms , i.e., bijective mappings between metric graphs (even those withdifferent edge sets!) that preserve distances, define an equivalence relation ≈ on the class ofall metric graphs which satisfy Assumption 2.2. If two graphs are isometrically isomorphic toeach other, then we are in one or both of the following situations:(1) the edge and vertex sets of one are a permutation (i.e. relabelling) of the edge andvertex sets of the other;(2) the graphs differ by the presence of dummy vertices.See also [KS02, Definition 5] and the discussion around it. We will need to make the followingdefinition for technical purposes, which will be necessary for the constructions in the comingsections up to and including Section 3. Definition 2.3. (1) We call any equivalence class of metric graphs satisfying Assump-tion 2.2 with respect to ≈ , an ur-graph .(2) If G is an ur-graph, then its canonical representative is the metric graph representative of G which has no vertices of degree two (or, if G is a loop, then its canonical representativeis any representative with exactly one vertex of degree two).(3) We will call the underlying discrete graph of the canonical representative of an ur-graph G the underlying discrete ur-graph of G (or discrete ur-graph associated with G ).In practice, we will not distinguish between different representatives of the same ur-graph;indeed, for spectral analysis different representatives of an ur-graph are indistinguishable (seeRemark 2.4). We will tacitly tend to identify an ur-graph G with any of its representatives asconvenient, most commonly (but not always) its canonical representative. We will thus alsospeak of ur-graphs as being compact metric spaces, and as satisfying Assumption 2.2, etc.There is a canonical notion of (scalar-valued) continuous functions over G with respect tothe distance defined above, and we stress that this notion is invariant under taking differentrepresentatives of the same ur-graph.Similarly, the Lebesgue measure, defined edgewise, induces in a canonical way a measure on G , allowing us to define square integrable functions on G : L ( G ) := M e m ∈E L ( e m ) ≃ M M m =1 L ([ x m − , x m ]) ,C ( G ) := ( f ∈ M e m ∈E C ( e m ) : f ( x j ) = f ( x k ) =: f ( v n ) if x j , x k ∈ v n for some v n ∈ V ) . J. B. KENNEDY, P. KURASOV, C. L´ENA, AND D. MUGNOLO
In order to define Laplacian-type operators we will require the Sobolev spaces H ( G ) := { f ∈ C ( G ) ∩ M e m ∈E H ( e m ) } and, for a given distinguished set V D ⊂ V of vertices, H ( G ) ≡ H ( G ; V D ) := { f ∈ H ( G ) : f ( v n ) = 0 for all v n ∈ V D } . Given the sesquilinear form(2.3) a ( f, g ) = ˆ G f ′ · ¯ g ′ d x ≃ M X m =1 ˆ x m x m − f ′ · ¯ g ′ d x, f, g ∈ H ( G ) , the associated self-adjoint operator on L ( G ) is the Laplacian − ∆ = − d dx defined on thedomain of functions from L e m ∈E H ( e m ) satisfying continuity and Kirchhoff conditions (sum ofinward-pointing derivatives is zero) at every vertex. Such vertex conditions, which we will call natural , are also known as standard, free, or sometimes Neumann–Kirchhoff conditions. TheLaplacian with Dirichlet conditions on a subset V D and natural conditions at all other verticesis the operator on L ( G ) which is associated with the form a restricted to H ( G ; V D ).Due to the positivity of a and the compact embedding of H ( G ) in L ( G ), the Laplacian onthe connected, compact graph G with natural vertex conditions has a sequence of non-negativeeigenvalues, which we will denote by0 = µ ( G ) < µ ( G ) ≤ µ ( G ) ≤ . . . → ∞ , repeating them according to their (finite) multiplicities; the eigenfunction corresponding to µ = µ ( G ) is just the constant function.We shall do likewise for the eigenvalues of the Laplacian on G with some Dirichlet conditions:0 < λ ( G ; V D ) < λ ( G ; V D ) ≤ . . . → ∞ . In practice we will abbreviate these to λ k ( G ) or even just λ k , k ≥
1, if the vertex set and thegraph are clear from the context. These eigenvalues admit the usual minimax and maximincharacterisations; in particular, we have(2.4) λ ( G ) = λ ( G ; V D ) = inf ( a ( f, f ) k f k L ( G ) : 0 f ∈ H ( G ; V D ) ) , while(2.5) µ ( G ) = inf ( a ( f, f ) k f k L ( G ) : 0 f ∈ H ( G ) , ˆ G f d x = 0 ) . In both (2.4) and (2.5), the infima are achieved only by the respective eigenfunctions, which aresign-changing and may be multiple in (2.5), but are unique up to scalar multiples and non-zeroeverywhere in (2.4).
Remark 2.4.
Suppose G and G ′ are isometrically isomorphic to each other in the sense de-scribed above. Then the respective spaces L , C and H on the two graphs are also isometricallyisomorphic to each other. It follows in particular that the corresponding Laplacians with nat-ural vertex conditions are unitarily equivalent to each other, and the respective eigenvalues areequal: µ k ( G ) = µ k ( G ′ ) for all k ≥
1. Likewise, if we fix a set V D ( G ) ⊂ V ( G ) of vertices of G , andchoose G ′ in such a way that the image of each point in V D ( G ) under the isomorphism is also a THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 9 vertex of G ′ , so that we may write V D ( G ) ≃ V D ( G ′ ), then H ( G ; V D ( G )) and H ( G ′ ; V D ( G ′ )) arealso isometrically isomorphic. Thus the corresponding Dirichlet Laplacians are likewise unitar-ily equivalent, and λ k ( G ; V D ( G )) = λ k ( G ′ ; V D ( G ′ )) for all k ≥
1. In other words, the eigenvaluesand eigenfunctions may be associated with the corresponding ur-graph; and for our purposes,within an ur-graph, i.e., an equivalence class of isometrically isomorphic graphs, we may at anytime pick any representative, as convenient.Finally, we mention in passing fundamental inequalities for the eigenvalues λ ( G ) and µ ( G )originally due to Nicaise [Nic87], which we will require on several occasions throughout thepaper. Theorem 2.5 (Nicaise’ inequalities) . Let G be any finite, compact connected (ur-) graph. Then (2.6) λ ( G ) ≥ π |G| and µ ( G ) ≥ π |G| , where in the first case G is equipped with at least one Dirichlet vertex. Equality in eitherinequality implies that G is a path graph (interval) of length |G| , with a Dirichlet vertex atexactly one endpoint and a natural (Neumann) condition at the other in the first case, andnatural conditions at both endpoints in the second case.Proof. The inequalities may be found in [Nic87, Th´eor`eme 3.1]. For the characterisation ofequality, see for example [Fri05] (or also [KN14, Theorem 3] in the case of natural conditions). (cid:3)
We refer to [BK13, Kur20, Mug14] for more background details on the properties of metricgraphs and Laplacian-type differential operators on them; we also refer to [BKKM19] and thereferences therein for more details on the eigenvalues λ k ( G ), µ k ( G ) and their dependence onproperties of the graph G .2.2. A motivating example.
Our goal is to study cutting metric graphs into pieces forminga partition; we shall call these pieces clusters . We shall require later on that partitions havecertain “good” properties, but we need to discuss first what kinds of splittings are possible atall.This subject is not new. Both Cheeger-like splittings as introduced in [Nic87] and the investi-gations in [BBRS12] restrict to the case of cuts performed in the interior of edges. These kindsof partitions are referred to as proper in [BBRS12], where their interplay with nodal domains(studied e.g. in [GSW04, Ber08]) is discussed.Here we wish to consider essentially all possibilities for making cuts, in particular whencuts are made at vertices of degree at least 3, using a concrete example to motivate what wewill introduce subsequently. More precisely, we will consider the lasso graph G depicted inFigure 2.1, formed by three edges e , e , e , as shown. vw ze e e Figure 2.1.
The lasso G . As done in [BBRS12], we will refer to any partitions where the cuts are made only at interiorpoints of edges as proper. Figure 2.2 illustrates two different ways to split G into two clusters,i.e., to create a proper 2-partition. v ˜ v ˜ vw z v ˜ v ˜ v ˜ v ˜ v w z Figure 2.2.
Two proper 2-partitions of the lasso G : with separating points atthe dummy vertices ˜ v and ˜ v , ˜ v , respectively.Note that using our convention to consider ur-graphs, any cut at an interior point of anedge can be considered as a vertex cut: every such point on the original graph can be seenas a dummy degree two vertex, before cutting one should choose a representative (from theequivalence class) with degree two vertex at the point one wish to cut through.At any rate, we will refer to the points at which cuts are made as separating points ; thesewill be introduced more formally in Section 2.3.Proper partitions are relatively easy to deal with, but are rather restrictive. Any sort offunctional we define on partitions should depend continuously on the points at which we arecutting, therefore we necessarily have to consider the limits that may arise when the cut reachesa vertex of degree ≥ • starting with the partition on the left of Figure 2.2 and letting the separating point ˜ v tend towards v , the limit partition should be the one depicted in Figure 2.3. • starting with the partition on the right Figure 2.2 and letting ˜ v , ˜ v tend towards v thelimit partition should coincide with the one depicted in Figure 2.4.The edge sets within each cluster are the same in both partitions; the way endpoints of theseedges are organised into vertices is different. These partitions can be obtained by cutting thelasso graph through vertex v in two different ways: separating the corresponding equivalenceclass of end points into two and three subclasses respectively.Partitions of the first type (corresponding to Figure 2.3) inherit all possible connections fromthe original graph and reflect its topology as closely as possible; for this reason we will refer tothem as faithful .We will call any other partition where we are still only altering the connectivity of our clustersat separating points rigid ; this is, in particular, the case of the partition in Figure 2.4 (thoughit is also true of the faithful partition from Figure 2.3. vvw z Figure 2.3.
A faithful 2-partition of the lasso G ; the only separating point is v . THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 11 w v vv ze e e Figure 2.4.
A rigid 2-partition of the lasso G ; again, the only separating point is v .Rigid partitions may appear less natural than faithful ones. As charming the notion of faithfulpartition may look at a first glance, it turns out that it is of little use: we will elaborate onthis in Section 3. Indeed, the point of departure of this article involves introducing a suitable,arguably natural metric on the space of graph partitions with respect to which neither the setof proper partitions, nor the set of faithful ones, is closed; however, the set of rigid partitionsis. This decisive topological feature is the main reason why we believe it is natural to considerthem.We conclude by considering a further relaxation, which also explains the use of the term rigid : namely, we may allow cuts not only at the points separating clusters but also at interiorpoints of clusters (note that these points are not necessarily interior points on some edges:these points could be vertices lying inside clusters), as long as each cluster stays connected: weshall refer to this kind of partition as loose . w v v zze e e Figure 2.5.
A loose 2-partition of the lasso G ; in this case, the only separatingpoint is v but we are additionally cutting through z .Loose partitions also define a closed set of partitions with respect to the natural metric weare going to introduce in Section 3.Finally, we regard partitions consisting of non-connected clusters as invalid. w v vv zze e e Figure 2.6.
An invalid 2-partition of the lasso G ; cutting through both v and z has led to a disconnected cluster. This is a faithful 3-partition, though.2.3. Graph partitions.
After informally sketching the ideas that motivate our classification,let us now introduce our notion of partition more precisely. We first need to recall an operationtransforming a graph into another one by joining or cutting through its vertices (cf., e.g., [BKKM19, Definitions 3.1 and 3.2]). We phrase this slightly differently, using the formalismintroduced in Section 2.1 and especially Definition 2.3.
Definition 2.6.
Let G , e G be ur-graphs. Then e G is called a cut of G if there exist a representative G ′ of G and a representative e G ′ of e G with vertex sets V ( G ′ ) = { v ( G ) , . . . , v N ( G ) } and V ( e G ′ ) = { ˜ v ( G ′ ) , . . . , ˜ v e N ( G ′ ) } and edge sets E ( G ′ ) and E ( e G ′ ), respectively, such that(a) E ( e G ′ ) = E ( G ′ ),(b) e N ≥ N , and(c) for all ˜ n = 1 , . . . , e N , in the notation and identification of Section 2.1, we have e V ˜ n ( e G ′ ) ⊂ V n ( G ′ )for some n = 1 , . . . , N .In words, the graph e G is formed from G by first picking a collection of vertices, in generalincluding dummy vertices in the interior of edges, of G (this is the choice G ′ ), and then cuttingthrough each such vertex v n of G ′ by removing adjacency relations to create new vertices ˜ v ˜ n out of v n . In practice, however, we will tend to suppress the tildes wherever feasible. As statedearlier, we will also tend not to distinguish between the ur-graph G and its representative G ′ ;in particular, in a slight abuse of notation, we will regard the vertices v , . . . , v N of G ′ as beingvertices of G . We also stress that we do not require cutting through vertices to produce a connected metric graph e G .Let us make this clearer by considering what happens if we only cut G at a single vertex. Definition 2.7.
Given two ur-graphs G , e G , keep the setup and notation of Definition 2.6.(1) Suppose there exist a representative G ′ of G , a representative e G ′ of e G , and(a) vertices v n ∈ V ( G ′ ) and ˜ v ˜ n , . . . , ˜ v ˜ n k ∈ V ( e G ′ ) such that V n = e V ˜ n ∪ . . . ∪ e V ˜ n k , and(b) there is equality e V ˜ n ( e G ′ ) = V n ( G ′ ) in condition (c) of Definition 2.6 for all ˜ n except ˜ n , . . . , ˜ n k .Then we say that e G has been obtained from G by cutting through the vertex v n (toobtain the vertices ˜ v ˜ n , . . . , ˜ v ˜ n k ). We call the vertices ˜ v ˜ n , . . . , ˜ v ˜ n k the image of thevertex v n under the cut.(2) We also say that the vertex v n in G corresponds to the vertices ˜ v ˜ n , . . . , ˜ v ˜ n k in e G , andthat G is obtained from e G by gluing the vertices ˜ v ˜ n , . . . , ˜ v ˜ n k to form v n .(3) We say that the vertex v n ( G ) has been cut nontrivially if k ≥ Definition 2.8 (Partitions of a graph) . Let k ≥ G be an ur-graph.(1) We call any set of k distinct connected metric graphs P := {G , . . . , G k } a k -partition of G if there is a cut e G = F k j =1 G i j of G , k ≥ k , whose connected compo-nents include G , . . . , G k , i.e., G i j = G j for all j = 1 , . . . , k , where i j = i j for j = j .In this case, we refer to the components G , . . . , G k as the clusters of the partition P ( arising from the cut e G ). THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 13 (2) If in (1) there exists a cut e G of G such that e G = F ki =1 G i , then we say the partition P = {G , . . . , G k } is exhaustive . With this definition, the k clusters are themselves compact metric graphs, which may beidentified with subsets of G . It will however often be useful to consider explicitly the subsetsof G which correspond to the clusters; to this end we make the following definition. Definition 2.9 (Cluster supports) . Let G be an ur-graph and let P = {G , . . . , G k } be a k -partition of G , arising from the cut e G = F k i =1 G i , k ≥ k , of G . We identify G and e G with anyrespective representatives satisfying the conditions of Definition 2.6, that is, in such a way that E ( G ) = E ( e G ).(1) For each i = 1 , . . . , k , we denote by Ω i the unique closed subset of G such that { e ∈ E ( e G ) : e ⊂ G i } = { e ∈ E ( G ) : e ⊂ Ω i } and call the set Ω i the cluster support (associated with the cluster G i ), or just support for short.(2) We call the set(2.7) Ω := k [ i =1 Ω i the support of the partition P .With this definition, the cluster supports Ω , . . . , Ω k are really a partition of G in the “classi-cal” sense; let us elaborate on this point. Indeed, we may think of the Ω i as the subsets of G outof which we form new graphs, the clusters G i , by cutting through vertices as desired. Thus, byconstruction, the Ω i are closed, connected subsets of G , and their interiors int Ω i , i = 1 , . . . , k ,are pairwise disjoint. Moreover, P is exhaustive if and only if the set Ω ⊂ G actually equals G .(For various practical reasons we are taking the cluster supports to be closed, not open, subsetsof G .)Finally, with the right choice of representative of G , we may suppose that, for each i =1 , . . . , k , we have Ω i = e i ∪ . . . ∪ e i Mi for some edges e i , . . . , e i Mi ∈ E ( G ). This means that ∂ Ω i ⊂ V ( G ) for all i = 1 , . . . , k ; and for each e ∈ E ( G ) there exists at most one i = 1 , . . . , k suchthat e ⊂ Ω i , exactly one if P is exhaustive. (We emphasise that ∂ Ω i is always the topologicalboundary of the closed set Ω i in the compact metric space G .)From now on, whenever P = {G , . . . , G k } is a k -partition of G , we will always use thenotation Ω , . . . , Ω k to denote the corresponding cluster supports, and Ω for the support of P (if distinct from G ), without further comment.Observe that if P is exhaustive and x ∈ ∂ Ω i for some i , then there must be at least one j = i such that x ∈ ∂ Ω i ∩ ∂ Ω j . Definition 2.10.
Let G be an ur-graph and let P = {G , . . . , G k } be a k -partition of G for some k ≥ v ∈ V ( G ) ∩ Ω a cut point (of P ) if there is no vertex˜ v ∈ k [ i =1 V ( G i ) Note that in the case of domains, some sources, such as [HHOT09], refer to exhaustive partitions as strong . such that v = ˜ v , that is, if v is nontrivially cut when constructing the partition. Werefer to the set C = C ( P ) ⊂ Ωof all cut points of P as the cut set of the partition P .(2) We will denote by(2.8) V D ( G i ) ⊂ G i the set of all vertices in G i which are obtained by nontrivially cutting through verticesof G ; in a slight abuse of terminology, we will also refer to its elements as cut points .(3) We call the separation set (of P ) the set(2.9) ∂ P := k [ i =1 ∂ Ω i ⊂ Ω . We refer to its elements as separating points .It follows from our definition of partitions that every separating point is a cut point, althoughthe converse need not be true; we also reiterate that we are assuming without loss of generality(by taking the right representative of the ur-graph) that each cut point, and in particular eachseparating point, is a vertex. Both the cut and the separation sets are clearly always finite.
Definition 2.11.
Let P = {G , . . . , G k } be a k -partition of G , and denote by Ω , . . . , Ω k therespective cluster supports. We say that Ω i , Ω j , i, j = 1 , . . . , k , i = j , are neighbours if ∂ Ω i ∩ ∂ Ω j = ∅ . In this case, we will also loosely refer to the corresponding clusters G i and G j asneighbours. Similarly, given a cut point v ∈ ∂ P , we will refer to each Ω i such that v ∈ ∂ Ω i asa neighbouring support of v .It turns out that there are several different, reasonably natural possibilities for defining classesof partitions of a metric graphs, as we intimated in Section 2.2. We stress that exhaustivityof a partition, in the sense of Definition 2.8(2), is not related to the following classification:exhaustivity does not imply, nor is it implied by, any of the following properties. Definition 2.12 (Classification of partitions) . Let G be an ur-graph.(1) Any partition P of G satisfying Definition 2.8 will be called loose .(2) A loose partition P of G will be called rigid if its cut and separation sets agree, that is,we only cut vertices on the boundary of Ω i to create the graph G i .(3) A partition P of G will be called faithful if it is rigid and additionally whenever aseparating point v lies in the cluster support Ω i , then in the corresponding cluster G i the image of v under the cut e G is incident with all edges e that were incident with v in G , such that e also lies in G i .(4) A partition P of G will be called internally connected if it is rigid and int Ω i = Ω i \ ∂ Ω i is connected, equivalently, if G i \ V D ( G i ) is connected, for all i = 1 , . . . , k .(5) A partition P of G will be called proper if it is rigid and all separating points are verticesof degree two in G .By definition, the cut and separation sets are allowed to be different only in a loose partition.It is clear from the definitions that every proper partition is faithful and internally connected,every faithful and every internally connected partition is rigid, and every rigid partition isloose, but the converse statements do not hold. For example, if G is a graph divided into THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 15 cluster supports Ω , . . . , Ω k , then any choice of spanning metric trees G , . . . , G k of these clustersupports determines a further loose partition. Example 2.13.
In the case of the lasso graph discussed in Section 2.2, we may choose to split G into the cluster supports Ω = e (interval) and Ω = e ∪ e (loop), so that ∂ P = ∂ Ω = ∂ Ω = { v } . Suppose we wish P to be exhaustive: in order to determine it, we need to specifythe clusters G and G : while a cluster G is uniquely determined by Ω , namely, it is the edge e , for the cluster G there are two possible choices, depicted in Figures 2.3 and 2.4, which leadto a faithful and a non-faithful but rigid 2-partition, respectively; both are internally connected.The third choice, of Figure 2.5, where to produce G we also cut through z , gives rise to a loose2-partition.If we allow P to be non-exhaustive and, say, take P = {G } , then P is faithful and internallyconnected (but still not proper). In this case, what happens to the set Ω under any cut givingrise to P is irrelevant for the classification of P . Example 2.14.
In the case of metric trees, our classification of partitions from Definition 2.12boils down to three cases.Cutting through a vertex of degree two creates by definition a proper 2-partition.Cutting through a single vertex v of degree deg v > k connected componentsfor any 2 ≤ k ≤ deg v ; the associated k -partition P that arises in this way is necessarilyexhaustive. More interestingly, if k = deg v , then P is both internally connected and faithful.If on the other hand k < deg v , then P is not internally connected (for there is some clustersuch that at least two different edges lie “on different sides” of the separating point v ); it isfaithful though, because by definition each cluster must be a connected metric graph in its ownright, hence no further cut can be made through v in any of the clusters.In particular, all loose partitions of metric trees are necessarily faithful, but there are rigidpartitions that are not internally connected.We will be primarily interested in the classes of loose and rigid partitions, and in exhaustivepartitions. For a fixed ur-graph G and k ≥
1, we denote the class of all exhaustive loose k -partitions of G by P k ( G ), or simply by P k if the graph G is clear from the context, the set ofall exhaustive rigid k -partitions of G by R k ( G ) or R k , and(2.10) P = P ( G ) := ∞ [ k =1 P k , R = R ( G ) := ∞ [ k =1 R k , the set of all loose exhaustive, and all rigid exhaustive, partitions of G , respectively.Finally, if Ω , . . . , Ω k ⊂ G are closed subsets of G with pairwise disjoint interiors, then foreach i = 1 , . . . , k , we set(2.11) ρ Ω i to be the finite set of all possible clusters G i that have Ω i as a cluster support and such thatthe partition P = {G , . . . , G k } is rigid . Note that ρ Ω i = ∅ ; indeed, ρ Ω i always contains exactlyone cluster corresponding to a faithful partition of P . We may also loosely refer to the clustersof such a partition as rigid clusters; we will do likewise for loose, faithful, internally connectedand proper clusters. Observe that as long as proper partitions are considered, there is no suchambiguity: each cluster support uniquely determines a cluster; in particular, the set ρ Ω i alwayscontains a single element. A spanning metric tree of a metric graph G is, by definition, a tree e G which is a cut of G . Example 2.15.
Returning again to the lasso graph discussed in Section 2.2, given the clustersupports Ω = e (interval) and Ω = e ∪ e (loop), we have that ρ Ω consists of a singleelement, the graph given by the edge e , while the set ρ Ω constains two graphs: an intervaland a loop, see Figures 2.3 and 2.4.3. Topological issues of graph partitions
Here we wish to construct a suitable topology on spaces of partitions, which will allow us togive existence results for minimisers of suitable functionals. Throughout this section, we will only work with exhaustive partitions, as these are more suited to topologisation and they willbe of primary interest in the sequel.3.1.
Primitive partitions.Definition 3.1.
Let G be an ur-graph and let P = {G (1)1 , . . . , G (1) k } and P = {G (2)1 , . . . , G (2) k } be two exhaustive, loose k -partitions of G . Then we say that P and P are similar , or sharea common cut pattern (of G ), if, up to the correct choice of representative of the ur-graph G and numbering of the clusters, for each i = 1 , . . . , k the clusters G (1) i and G (2) i have the sameunderlying discrete graph (see Definition 2.1), and there is a bijection between the cut sets C ( P ) , C ( P ) (see Definition 2.10). Proposition 3.2.
Suppose G is a fixed ur-graph and let k ≥ .(1) Similarity between k -partitions of G , as in Definition 3.1, is an equivalence relation. Itdivides P k ( G ) into a finite number of cells.(2) If two exhaustive, loose k -partitions P , e P of G are similar, then after renumbering theclusters if necessary, for each i = 1 , . . . , k , G (2) i can be obtained from G (1) i by lengthening orshortening edges of G (1) i .(3) If two exhaustive, loose k -partitions P , e P of G are similar and P is rigid (respectively,faithful, internally connected or proper), then so too is e P .Proof. (1) is immediate, since all properties of similarity may be characterised in terms ofbijections.(2) It suffices to prove that the same is true of any two metric graphs G and G which havethe same underlying discrete graph G . But this, in turn, is an immediate consequence of thedefinition (Definition 2.1): the edges e (1)1 , . . . , e (1) M of G and e (2)1 , . . . , e (2) M of G are in a canonicalbijection to each other, both being in bijective correspondence with the edges e , . . . , e M of G ;moreover, this bijective correspondence preserves all adjacency and incidence relations. Hence,if for each i = 1 , . . . , k we replace the edge e (1) i with an edge of length | e (2) i | , then the resultinggraph is isometrically isomorphic to G .(3) follows since cut patterns completely describe the connectivity of the resulting clustersin the neighbourhood of any cut point. (cid:3) Definition 3.3.
We call the equivalence classes with respect to the above equivalence relation primitive k -partitions . We will denote the primitive partition associated with the cut pattern C by T = T C ⊂ P k . Since by definition P k consists of exhaustive partitions, any primitive k -partition is neces-sarily exhaustive. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 17
Partition convergence.
The equivalence discussed in the previous subsection gives riseto a notion of convergence of partitions within each primitive partition, which we now wish tointroduce. To begin with, we need the notion of convergence of a sequence of graphs havingthe same underlying discrete topology, similar to what was considered in [BL17, § G = ( V , E ), let Γ G be the set of all ur-graphs whose underlyingdiscrete graph is G , in the sense of Definition 2.1. We assume here and throughout that theindexing of the edges and vertices is consistent, in the sense that if E = { e , . . . , e M } and G (1) , G (2) ∈ Γ G , then up to the correct choice of representatives of the ur-graphs we have E ( G ( n ) ) = { e ( n )1 , . . . , e ( n ) M } and the bijection Ψ i : E → E maps e i to e ( n ) i for all i = 1 , . . . , M ,with corresponding statements for the vertices, n = 1 ,
2. Observe that each
G ∈ Γ G is uniquelydetermined by its vector ( | e | ) e ∈E of edge lengths, hence we can define(3.1) d Γ G ( G , e G ) := d R M (cid:0) ( | e | ) e ∈E , ( | ˜ e | ) ˜ e ∈ ˜ E (cid:1) , G , e G ∈ Γ G , where d R M is the Euclidean distance on R M . Proposition 3.4.
Given a discrete graph G , (Γ G , d Γ G ) is a separable metric space with respectto the Euclidean distance in R M . This metric structure induces the same topology as the one discussed in [BBRS12, §
2] andthe one used in [BL17].We can now consider Cauchy sequences G ( n ) in Γ G ; however, they need not converge inΓ G , since one or more edge lengths may tend to 0. We can however consider the canonicalcompletion Γ G of Γ G : it consists of equivalence classes of Cauchy sequences of metric graphsin Γ G with respect to the equivalence relation of having distance d Γ G ( G ( n ) , e G ( n ) ) vanishing as n → ∞ . One can identify Γ G with the simplex of all vectors in the positive orthant of R M whose size agrees with the total length of G , i.e.,Γ G ≃ ( ( x , . . . , x M ) : x i ≥ M X i =1 x i = M X i =1 | e i | ) . The limit of a converging sequence ( G ( n ) ) n ∈ N ⊂ Γ G may hence be identified with an ur-graph G ( ∞ ) whose edge lengths are the (possibly vanishing) limits of the edge lengths of the approxi-mating graphs G ( n ) ; accordingly G ( ∞ ) may well have a different underlying discrete graph witha lower number of vertices and edges; and it may contain loops and parallel edges even if theapproximating graphs do not. We may group the vertices of G ( n ) according to the rule v ( n ) , w ( n ) ∈ V ( G ( n ) ) are equivalent if and only if dist G ( n ) ( v ( n ) , w ( n ) ) n →∞ −→ v ( ∞ ) of G ( ∞ ) is associated a unique equivalence class of vertices of G ( n ) of this form, which we will denote by [ v ( ∞ ) ].Let us explicitly formulate the following useful observations. Lemma 3.5.
Let ( G ( n ) ) n ∈ N converge to G ( ∞ ) in Γ G . Then(1) the total length |G ( n ) | tends to |G ( ∞ ) | ;(2) G ( ∞ ) is connected provided the G ( n ) are. We also note for future reference that the Laplacian eigenvalues we are considering, in-troduced in Section 2.1, behave well with respect to this notion of convergence. Here thecorrespondence between vertices is necessary to identify the correct limiting vertex conditions.
Lemma 3.6.
Let ( G ( n ) ) n ∈ N converge to G ( ∞ ) = ∅ in Γ G . Then(1) µ ( G ( n ) ) → µ ( G ( ∞ ) ) ;(2) if a vertex set V D in the underlying discrete graph G is chosen and Dirichlet conditionsare applied at all vertices in G ( n ) corresponding to V D , and if Dirichlet conditions areapplied at exactly those vertices v of G ( ∞ ) such that at least one vertex in [ v ] correspondsto a vertex in V D , then λ ( G ( n ) ) → λ ( G ( ∞ ) ) .If G ( n ) → ∅ , then µ ( G ( n ) ) → ∞ as n → ∞ . If in addition V D ( G ( n ) ) = ∅ , then also λ ( G ( n ) ) →∞ .Proof. (1) follows from the method described in [BL17, Appendix A] (which can also be easilyadapted to (2)); alternatively, see [BLS19] for a more detailed treatment of both. The casewhere no edge lengths converge to zero is already covered in [BK13, § µ ( G ( n ) ) ≥ π / |G ( n ) | → ∞ and λ ( G ( n ) ) ≥ π / |G ( n ) | → ∞ (in the latter case as long as atleast one Dirichlet vertex is present). (cid:3) With this background, we can now return to partitions and in particular define the notionof convergence of a sequence of partitions. For the rest of the section, we assume that G is afixed ur-graph satisfying (up to the correct choice of representative) Assumption 2.2, and fix aprimitive k -partition T of G ; we suppose that the clusters of each partition P = {G , . . . , G k } have the respective underlying discrete graphs G , . . . , G k (for a fixed order). Then, as above,setting E i := | E ( G i ) | to be the number of edges of G i , each G may be uniquely identified witha vector in R E i + ; this means that each P = ( G , . . . , G k ) may be identified with a vector in R E + × . . . × R E k + ≃ R E + whose (strictly) positive entries sum to the total length |G| of the graph G , that is, we have the identification(3.2) T ≃ Θ T := (cid:8) x = ( x , . . . , x E ) ∈ R E : x j > j and | x | = |G| (cid:9) , where E = k P i =1 E i = k P i =1 | E ( G i ) | and | x | is the 1-norm of the vector x . Now if two partitions P = {G , . . . , G k } and e P = { e G , . . . , e G k } are similar, P , e P ∈ T = T C (and in particular consist ofclusters that have the same underlying discrete graphs, say G , . . . , G k ), then we can introduce(3.3) d ( P , e P ) := k X i =1 d Γ G i ( G i , e G i ) , where d Γ G i is the distance introduced in equation 3.1. This distance induces an equivalenttopology to the one induced by the Euclidean distance between the points in the set Θ T corre-sponding to the respective partitions P and e P . The following result is immediate. Lemma 3.7.
Let C be a cut pattern of G . Then T = T C is a metric space with respect to thedistance introduced in (3.3) . In order to check the plausibility of this metrisation of the partition space, let us explicitlyrecord the following observation.
Proposition 3.8.
Suppose T is any primitive partition and ( P n ) n ∈ N ⊂ T is a sequence of k -partitions which is Cauchy with respect to the metric (3.3) . Then the limit partition P ∞ ∈ T isalso exhaustive. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 19
However, this metric space is non-complete, since given a Cauchy sequence ( P n ) n ∈ N ⊂ T itcannot be excluded that one or more clusters vanish in the limit, i.e., |G ( n ) i | →
0, leading toan m -partition of G with m < k as a limit object; these correspond to the limit points in theEuclidean set Θ T from (3.2) with one or more entries equal to zero.The spectral energies we will consider in the sequel will turn out to be continuous withrespect to this metric, cf. Lemmata 4.5 and 6.3. This is an immediate corollary of Lemma 3.6.However, if the P n are k -partitions and P n → P ∞ for some m -partition P ∞ with m < k , thenwe do not in general expect spectral continuity, since the corresponding partition energies willdiverge to ∞ (cf. the proof of Lemma 4.6).Nevertheless, as above, it is natural to consider the canonical completion T , which consistsof equivalence classes of Cauchy sequences of partitions with respect to the equivalence relationof having vanishing distance d ( P n , e P n ) in the limit, which corresponds to Θ T ⊂ R E . Lemma 3.9.
Let C be a cut pattern of G . Then T C is compact.Proof. This is immediate since T C may be identified with the closed and bounded subset Θ T of E -dimensional Euclidean space. (cid:3) More generally, if A ⊂ P k is any set of k -partitions, then A is the union of the sets A ∩ T overall primitive partitions T . Obviously, it is possible that a given m -partition P ∈ A may lie inthe closure of more than one primitive partition. Moreover, the sets P k and R k are themselvesnot closed, although, as we will see shortly, S i ≤ k P k and S i ≤ k R k are.It is also natural to ask which types of partition from our classification, Definition 2.12, areclosed in the metric (3.3). Example 3.10.
Let us review the proper 2-partitions of the lasso graph of Section 2.2. As theseparating point ˜ v wanders towards v in Figure 2.2, the corresponding partition P converges,with respect to the metric introduced in (3.3), towards the faithful (but non-proper) 2-partitionin Figure 2.3. On the other hand, as the ˜ v , ˜ v approach v in Figure ?? , the correspondingproper (and hence faithful) partition P converges towards the rigid, non-faithful 2-partition inFigure 2.4. Observe that the cut pattern and hence the underlying discrete graphs of these twolimiting partitions are different.Hence, neither the class of proper, nor faithful partitions is closed; nor is the class of internallyconnected partitions, as can be shown using Example 2.14. In particular, connectivity of theclusters, even if it holds for a sequence of partitions, can be destroyed in the limit. On theother hand, if ( P n ) n ∈ N ⊂ T is a sequence of loose partitions of a given primitive partition, thenthe limit object is clearly still a well-defined m -partition for some 1 ≤ m ≤ k ; in particular, itis loose, and thus S i ≤ k P k is closed. The following proposition establishes that a correspondingstatement holds for rigid partitions; and it is for this reason that we will tend to favour thesetwo partition classes over the respective classes of proper, faithful and internally connectedones. Proposition 3.11.
Suppose T is any primitive partition and ( P n ) n ∈ N ⊂ T is a sequence ofrigid k -partitions which is Cauchy with respect to the metric (3.3) . Then the limit partition P ∞ ∈ T is a rigid m -partition for some m , ≤ m ≤ k . In particular, S i ≤ k R k is closed. Proof.
Fix i = 1 , . . . , k . Now since obviously G ( n ) i → G ( ∞ ) i with respect to the metric ofequation (3.1), by Lemma 3.5 we have that G ( ∞ ) i is connected; in particular, P ∞ cannot havemore than k clusters. To check the rigidity condition, we also need to show that any vertex v ∈ int Ω ( ∞ ) i is not cut through in G ( ∞ ) i . So let v ∈ int Ω ( ∞ ) i be arbitrary, then we first observethat v ∈ int Ω ( n ) i for all sufficiently large n . Since P n was assumed rigid, any edge of G incidentwith v remains incident with v in G ( n ) i , and none of these edges in G ( n ) i has length convergingto zero. In particular, the incidence relations at v are preserved in the limit graph G ( ∞ ) i . Weconclude that P ∞ is rigid. (cid:3) Existence results for energy functionals.
In this section we prove a general existenceresult for extremisers of functionals Λ :
P 7→ R defined on certain sets of partitions.Since each primitive partition is a metric space by Lemma 3.7, as is the disjoint union of allprimitive partitions (up to allowing the distance function to attain the value + ∞ ), all usualtopological notions are well-defined: lower semicontinuity will play a key role in what follows. Definition 3.12.
Let A ⊂ P ( G ) be a set of exhaustive partitions. We say the functional J : A → R is(1) lower semi-continuous (lsc) if, whenever T is a primitive partition and ( P n ) n ∈ N ⊂ A ∩ T converges to some P ∈ A ∩ T , we have that J ( P ) ≤ lim inf n →∞ J ( P n );(2) strongly lower semi-continuous (slsc) if, whenever T is a primitive partition and ( P n ) n ∈ N ⊂ A ∩ T converges to some P ∈ A ∩ T , we have that J ( P ) ∈ R is well defined and J ( P ) ≤ lim inf n →∞ J ( P n ).(Strong) upper semi-continuity and (strong) continuity may be defined analogously. Note,however, that continuity of J is not assumed on the closure of its domain A ; in particular, evenif J is continuous on the whole of P or R , it need not be bounded from above or below, noteven on the set of all k -partitions, since we do not rule out discontinuities, or even divergence, J ( P n ) → ±∞ , if one or more clusters of P n disappear in the limit. (This will, for example, bethe case for the continuous functionals Λ Dk,p and Λ
Nk,p , see Lemmata 4.5 and 4.6.)
Theorem 3.13.
Let k ≥ and let A ⊂ P = P ( G ) with A ∩ P k = ∅ . Suppose that the functional J : A → R is strongly lower semi-continuous on A . Suppose in addition that at least one of thefollowing conditions holds:(1) J ( P n ) → ∞ whenever there exist clusters G n in P n ∈ A such that |G n | → as n → ∞ ;or(2) for every ℓ -partition P ( ℓ ) ∈ A , ℓ = 1 , . . . , k − , there exists an ( ℓ +1) -partition P ( ℓ +1) ∈ A such that J ( P ( ℓ +1) ) ≤ J ( P ( ℓ ) ) .Then there is at least one exhaustive k -partition P ∗ ∈ A ∩ P k realising (3.4) J ( P ∗ ) = inf { J ( P ) : P ∈ A ∩ P k } . If A ⊂ R , that is, if we restrict to rigid partitions, then there is at least one exhaustive rigid k -partition P ∗ satisfying (3.4) . If we assume A to be contained in the set of proper, or faithful, or internally connectedpartitions, then in general the minimiser P ∗ is merely rigid, since the former sets are notclosed. We will give concrete examples of this elsewhere; see for example Example 4.10 andalso Section 8.2, and cf. also Example 3.10. We emphasise that lower semi-continuity by itself THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 21 is not enough to guarantee the existence of a minimiser in A , since the lower semi-continuitycondition does not require J ( P n ) → J ( P ∞ ) if A is open and A ∋ P n → P ∞ ∈ ∂A , even if J ( P ∞ ) is actually well defined. We likewise need (1) or (2) to prevent the only limits of anyminimising sequences from being m -partitions for some m < k .While the monotonicity-like condition in (2) may seem a little unusual, it will be directlyapplicable to the partitions of max-min type considered in Section 6. Proof of Theorem 3.13.
Let ( P n ) n ≥ be a sequence of k -partitions in A ∩ P k such that J ( P n ) → inf A ∩ P k J ( P ) as n → ∞ . Since there are only finitely many primitive partitions, there mustexist a subsequence, which we shall still denote by ( P n ), such that the P n are all similar, P n ∈ T for some primitive partition T . We will write P n := {G ( n )1 , . . . , G ( n ) k } .By Lemma 3.9 there exists an m -partition P ∞ ∈ A ∩ T , m ≤ k , such that up to a subsequence P n → P ∞ . If A ⊂ R , that is, if all partitions under consideration are rigid, then since R isclosed by Proposition 3.11, also P ∞ ∈ R .To finish the proof, it suffices to show that P ∞ is actually a k -partition, since the stronglower semi-continuity of J already implies that(3.5) J ( P ∞ ) ≤ lim inf n →∞ J ( P n ) . Assume condition (1). Then since the sequence ( J ( P n )) is bounded from above, | Ω ( n ) i | cannotconverge to zero for any i , and hence, by Lemma 3.5, also |G ( ∞ ) i | > P ∗ = P ∞ .Instead assume condition (2). Suppose that the minimising partition P ∞ found above is an m -partition for some 1 ≤ m ≤ k . In this case, (2) still gives us (3.5), and then, upon sufficientlymany applications of the second part of (2) to P ∞ we obtain a k -partition e P with J ( e P ) ≤ J ( P ∞ ) = inf P∈ A ∩ P k J ( P ) , meaning we have found a minimal k -partition e P = P ∗ . If A ⊂ R , then e P ⊂ A ⊂ R byassumption. (cid:3) Existence of spectral minimal partitions
We can now introduce the first major types of spectral energy functionals we wish to consider.From now on, we will no longer need to distinguish between metric and ur-graphs, so we willalways suppress this technicality and assume that G is a fixed metric graph – say, the canonicalrepresentative of an ur-graph. We also fix k ≥ P = {G , . . . , G k } ∈ P k = P k ( G ) is a k -partition of G . On each of the graphs G i , i = 1 , . . . , k , we consider either:(1) the smallest nontrivial eigenvalue µ ( G i ) of the Laplacian with natural vertex conditions,given by (2.5); we have µ ( G i ) > G i is connected by definition; or(2) the smallest eigenvalue λ ( G i ) = λ ( G i ; V D ( G i )) > V D ( G i ), cf. (2.4) and (2.8).In either case, we associate a spectral energy with the graph G i , and thus, collating these overall i , with the partition P of G . There are multiple possible ways to do so; the particularproblems we shall consider in this section are as follows: for any given p ∈ (0 , ∞ ], we consider the energies(4.1) Λ Np ( P ) = (cid:18) k k P i =1 µ ( G i ) p (cid:19) /p if p ∈ (0 , ∞ ) , max i =1 ,...,k µ ( G i ) if p = ∞ , and(4.2) Λ Dp ( P ) = (cid:18) k k P i =1 λ ( G i ) p (cid:19) /p if p ∈ (0 , ∞ ) , max i =1 ,...,k λ ( G i ) if p = ∞ , in each case for a given k -partition P = {G , . . . , G k } ∈ P k of a graph G . Here we have written,and we will always understand, λ ( G i ) = λ ( G i ; V D ( G i )) , where we will always take the set of Dirichlet vertices of G i to be the set V D ( G i ) of cut pointsin G i , as defined in Definition 2.10 (indeed, this motivates the notation V D ); likewise, we willwrite H ( G i ) in place of H ( G i ; V D ( G i )). The problem is then to minimise the energies (4.1)and (4.2), respectively, that is, to solve forinf P∈ A ∩ P k Λ Np ( P ) and inf P∈ A ∩ P k Λ Dp ( P )for a suitable set or class of partitions A . There are multiple reasonably natural possiblechoices for the set A over which we can seek the infimum, in particular, we may consider anyof the classes listed in Definition 2.12. Iin keeping with the usual convention when dealingwith domains we will be mostly interested in exhaustive partitions, although the problems weconsider would also be well posed without this restriction.For example, we recall that in [BBRS12] the authors were interested in proper (and ex-haustive) partitions, and in particular thoroughly studied the local minima of Λ D ∞ and theirgeometric properties (see especially [BBRS12, Theorems 2.7 and 2.10]); however, the actualquestion of existence of minimisers was not discussed (whether within the class of proper par-titions or in general). We also recall that the internally connected partitions are exactly thoserigid partitions for which G i remains connected after removing the sets V D ( G i ) ≃ ∂ Ω i , makingthem an a priori natural class of partitions on which to consider Dirichlet problems. However,as noted in Section 3, these classes are not closed in the natural partition topology; hence wecannot expect to find a minimiser within the respective classes (see Example 4.10 below) – eventhough the energies (4.1) and (4.2) are continuous, as we will show shortly. Based on the factthat the classes of rigid and loose partitions are closed, they will be of primary interest for us,that is, for p ∈ (0 , ∞ ] we will principally consider the four problems of finding(4.3) L Nk,p = L Nk,p ( G ) := inf P∈ R k Λ Np ( P ) , L Dk,p = L Dk,p ( G ) := inf P∈ R k Λ Dp ( P ) , e L Nk,p = e L Nk,p ( G ) := inf P∈ P k Λ Np ( P ) , e L Dk,p = e L Dk,p ( G ) := inf P∈ P k Λ Dp ( P ) , THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 23 where we recall that P k and R k are the sets of all loose and rigid exhaustive k -partitions of G ,respectively. We will refer to these partition problems as Neumann (or natural ) and
Dirichlet problems.
Example 4.1.
Let G be an equilateral pumpkin graph on 3 edges of, say, length 1. Then it iseasy to check (cf. Lemma 8.1 or Example 8.3) that L N ,p = π for all p ∈ (0 , ∞ ]: this value isattained by the partition P in Figure 4.1, which is also unique up to isomorphism. (This rigidpartition is also minimal among the loose ones, i.e., Λ Np ( P ) = L N ,p = e L N ,p for all p ∈ (0 , ∞ ].) Figure 4.1.
A minimal Neumann 2-partition of a pumpkin on 3 edgesThe principal goal of this section is to show that such minimal partitions always exist, andindeed for all four problems listed above.
Remark 4.2.
In the case of domains, only the Dirichlet, not the Neumann problem, has beenstudied, since unlike on metric graphs, on domains the latter minimisation problem is not welldefined. Let us expand on this point a little. Suppose for simplicity that Ω ⊂ R is a smoothdomain and consider k -partitions P = { Ω , . . . , Ω k } of Ω for some fixed k ≥
1. In the easiercase of non-exhaustive partitions, we take ω ε to be a dumbbell consisting of two disks of radius d = d ( k, Ω) > ε >
0, where theparameters are chosen in such a way that Ω constains at least k disjoint copies of ω ε , for any ε > P ε is then the partition consisting of k such copies of ω ε , then since µ ( ω ε ) → ε → p ∈ (0 , ∞ ], 0 ≤ inf P Λ Np ( P ) ≤ lim ε → Λ Np ( P ε ) = 0 , meaning that the minimisation problem is not well defined. If we restrict to exhaustive parti-tions, then the same principle and conclusion apply, but the construction is harder to describe ingeneral. Instead, we illustrate the idea with a figure (Figure 4.2) sketching such a constructionfor 2-partitions of the square.Let us next give a few basic properties of the problems (4.3). It is immediate that L Nk,p ≥ e L Nk,p and L Dk,p ≥ e L Dk,p , since R k ⊂ P k . Actually, before proceeding let us note that the only quantityof interest in the Dirichlet case is L Dk,p : Lemma 4.3.
For any graph G , any k ≥ and any p ∈ (0 , ∞ ] , we have L Dk,p = e L Dk,p .Proof.
We only have to prove “ ≤ ”. Let e P ∈ P k ; we will construct a partition P ∈ R k whichhas the same cluster supports and whose every cluster has an eigenvalue which is no larger thanthe eigenvalue of the corresponding cluster of e P . In fact, if e G i has support Ω i and G i ∈ ρ Ω i isany rigid cluster with the same cluster support, then H ( e G i ) may be identified with a subspace ε ε Figure 4.2.
A sequence of exhaustive 2-partitions of the unit square: one par-tition element is white, the other grey. The thin joining passages are of width ε >
0; it can be shown that as ε →
0, the corresponding Neumann partitionenergy also converges to 0.of H ( G i ) since the zero condition can only be imposed at more points of e G i than of G i . Itfollows from the variational characterisation (2.4) that λ ( e G i ) ≥ λ ( G i ) for all i . The claim nowfollows. (cid:3) Remark 4.4.
Let us explicitly stress that if P = {G , . . . , G k } ∈ R k is a rigid partition(hence, cut points and separation points agree) and we are interested in Λ Dp ( P ), then λ ( G i ) isindependent of the choice of the graph G i ∈ ρ Ω i associated with Ω i , i = 1 , . . . , k , as long as thisis made in accordance with Definition 2.8. In other words, λ ( G i ) is independent of G i ∈ ρ Ω i .This is because a Dirichlet condition is imposed at all cut/separation points anyway; thus, itdoes not matter whether (or how) these vertices are joined in G i . Hence, in these cases, onemay ignore the distinction between the cluster supports Ω i and the clusters G i (in particular,minimising over the class of faithful partitions is the same as minising over rigid partitions forthe Dirichlet problem). In practice, we will always do this, that is, when considering (only)Dirichlet partition problems (among all rigid partitions) we will not distinguish between clustersand their supports .We next establish that both our spectral energy functionals (4.1) and (4.2) are indeed con-tinuous with respect to the notion of partition convergence introduced in Section 3.2. Lemma 4.5.
Suppose P n and P ∞ are exhaustive loose k -partitions of a graph G such thatthe P n are similar, and P n → P ∞ with respect to the metric of (3.3) . Then, for any given p ∈ (0 , ∞ ] , Λ Np ( P n ) → Λ Np ( P ∞ ) and Λ Dp ( P n ) → Λ Dp ( P ∞ ) as n → ∞ .Proof. This follows immediately from the definitions of Λ Np and Λ Dp and Lemma 3.6. (cid:3) Actually, we can say more.
Lemma 4.6.
The functionals Λ Np and Λ Dp are strongly lower semi-continuous on P k ( G ) (seeDefinition 3.12(2)), for any given p ∈ (0 , ∞ ] .Proof. If P n ∈ P k are k -partitions of a graph G and P n → P ∞ for some m -partition P ∞ with m < k , then there exists at least one sequence of clusters, say G ( n ) i , whose total length |G ( n ) i | →
0. Nicaise’ inequalities (Theorem 2.5) yield(4.4) µ ( G ( n ) i ) ≥ π |G ( n ) i | = π | Ω ( n ) i | and λ ( G ( n ) i ) ≥ π |G ( n ) i | = π | Ω ( n ) i | THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 25 for all i = 1 , . . . , k and all n ≥
1, whence Λ Np ( P n ) , Λ Dp ( P n ) → ∞ as n → ∞ , for any p ∈ (0 , ∞ ].The strong lower semi-continuity follows since Λ Np ( P ∞ ) , Λ Dp ( P ∞ ) < ∞ , when combined withthe result of Lemma 4.5. (cid:3) Given a graph G , we can now prove the existence of k -partitions achieving the infimal values L Nk,p ( G ) and L Dk,p ( G ) (and hence e L Dk,p ( G )), as well as e L Nk,p ( G ). Theorem 4.7.
Fix k ≥ and p ∈ (0 , ∞ ] and let A ⊂ P k be any set of k -partitions. Thenthere exist k -partitions P N , P D ∈ A ∩ P k of G such that Λ Np ( P N ) = inf A ⊂ P k Λ Np ( P ) and Λ Dp ( P D ) = inf A ⊂ P k Λ Dp ( P ) . In particular: if A ⊂ R k is a set of rigid k -partitions, then P N and P D are also rigid k -partitions, respectively. We will see in Section 7.1 that for fixed k ≥ p . Proof.
It suffices to show that the functionals Λ Np and Λ Dp , which are both defined on P ⊃ A ,satisfy the conditions of Theorem 3.13(1). Strong lower semi-continuity was established inLemma 4.6, while condition (1) follows immediately from (4.4). (cid:3) Corollary 4.8.
Fix k ≥ and p ∈ (0 , ∞ ] . Then there exist a loose k -partition e P N ∈ P k andrigid k -partitions P N , P D ∈ R k such that Λ Np ( e P N ) = e L Nk,p ( G ) , Λ Np ( P N ) = L Nk,p ( G ) , and Λ Dp ( P D ) = L Dk,p ( G ) = e L Dk,p ( G ) . Remark 4.9.
If, for any k ≥ p ∈ (0 , ∞ ], e P N = {G , . . . , G k } is a loose k -partitionachieving the minimum for L Nk,p ( G ), then we may always assume without loss of generality thatthe clusters G , . . . , G k are all trees. This is because cutting through any vertices in G i can onlydecrease µ ( G i ) (see, e.g., [BKKM19, Theorem 3.10(1)]). We will see this principle in action inExample 8.4 below.We next give a simple example to show that the minimal partition realising L Dk,p ( G ) withinthe class of all internally connected exhaustive partitions need not be internally connected andexhaustive, even if it can be approximated by such partitions. Example 4.10.
Let G be a star graph consisting of three edges e , e , e , each of lengthone, attached at a common vertex v . Then there is an optimal internally connected but non-exhaustive 2-partition, for both L Dk, ∞ and L Nk, ∞ , given by, say, P ∗ = {G , G } , with G = e , G = e .If, however, we search for rigid and exhaustive minimal partitions, then up to permutationof the edges, their cluster supports must all have the form Ω = e and Ω = e ∪ e , seeExample 2.14. In the Dirichlet case, since ∂ P ∗ = { v } and removing v disconnects Ω , anyrigid minimiser is not internally connected. A similar principle holds if we search for an optimal k -partition of a star on n equal rays e , . . . , e n , with n > k .We observe in passing that in all these cases the non-exhaustive partition achieving L Dk, ∞ ( G )is nodal , while the exhaustive partitions are generalised nodal , see Definitions 5.2 and 5.3 below. Remark 4.11. (1) For any graph G and any given p ∈ (0 , ∞ ] and 1 ≤ k ≤ k , we have themonotonicity statements(4.5) L Dk ,p ( G ) ≤ L Dk ,p ( G ) and(4.6) e L Nk ,p ( G ) ≤ e L Nk ,p ( G ) . The argument is the same in both cases, so we restrict ourselves to the Dirichlet case: we suppose P = {G , . . . , G k } is an exhaustive rigid k -partition realising L Dk ,p ( G ), and for simplicity wetake k = k −
1. Suppose without loss of generality that G k − and G k are neighbours (seeDefinition 2.11), and that λ ( G k ) = max i =1 ,...,k λ ( G i ). We glue these two clusters together:more precisely, we define e G k to be the rigid cluster, unique in the sense of Remark 4.4, whosesupport is exactly Ω k ∪ Ω k ; we also set e G i := G i for all i = 1 , . . . , k . Then it is easy tocheck that e P := { e G , . . . , e G k } is an exhaustive rigid k -partition of G . Moreover, by eigenvaluemonotonicity with respect to graph inclusion, λ ( e G k ) ≤ min { λ ( G k − ) , λ ( G k ) } = λ ( e G k − ).Hence, for any p ∈ (0 , ∞ ),1 k k X i =1 λ ( G i ) p ≥ k k − X i =1 λ ( e G i ) p + 1 k max i =1 ,...,k λ ( e G i ) p = 1 k − k X i =1 λ ( e G i ) p − k ( k − k X i =1 λ ( e G i ) p + 1 k max i =1 ,...,k λ ( e G i ) p ≥ k − k X i =1 λ ( e G i ) p − k − k ( k −
1) max i =1 ,...,k λ ( e G i ) p + 1 k max i =1 ,...,k λ ( e G i ) p = Λ Dk,p ( e P ) p . Since the inequality Λ D ∞ ( P ) ≥ Λ D ∞ ( e P ) is immediate, we obtain Λ Dp ( P ) ≥ Λ Dp ( e P ) for all p ∈ (0 , ∞ ]. This yields (4.5). For e L N we use Remark 4.9 to guarantee that without loss of generalitythe clusters are all trees; now [BKKM19, Theorem 3.10(1)] gives the monotonicity when gluing G k − and G k together, which may be done at a single vertex.(2) Note that inequality in (1) need not always be strict: for p = ∞ , if G is the equilateralstar graph from Example 4.10 (see also Examples 2.14 and 7.2), then L D , ∞ ( G ) = L D , ∞ ( G ) = π , where in the notation of Example 4.10 the optimal 3-partition is given by Ω i = e i for i =1 , ,
3. Inequality need not be strict even if the corresponding minimal partitions are internallyconnected; an example is the graph considered in Proposition 5.17: we will show there thatfor this graph we even have L D , ∞ = L D , ∞ despite there being internally connected partitionsrealising both minima.However, although for each k there is a rigid k -partition achieving L Nk,p ( G ), it is not actuallyclear whether the monotonicity property analogous to (4.5) holds: a necessary condition is thefollowing seemingly obvious conjecture, which will also play a role in Section 6. Conjecture 4.12.
Suppose G is a finite, compact, connected metric graph. Then there existsan exhaustive rigid 2-partition P = {G , G } of G such that µ ( G ) ≤ min { µ ( G ) , µ ( G ) } . In fact, if Conjecture 4.12 is not true, then there is a graph G such that µ ( G ) = L N ,p ( G ) > Λ Np ( P ) for every exhaustive rigid 2-partition P of G and any p ∈ (0 , ∞ ], and hence L N ,p ( G ) > THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 27 L N ,p ( G ). However, it is true for a large class of graphs, as the following observation shows. Thiswill also be used in the proof of Theorem 6.4 below. Lemma 4.13.
Conjecture 4.12 is true whenever G has a bridge , that is, an edge or a vertexwhose removal disconnects G . In particular, it is true for trees.Proof. Suppose that removing v ∈ G disconnects G into two subsets Ω and Ω whose intersec-tion is only { v } , and let G ∈ ρ Ω and G ∈ ρ Ω be the clusters of maximal connectivity, forwhich no further cuts are made at v , i.e., such that every other graph in ρ Ω i is a cut of G i , i = 1 , µ ( G ) ≤ µ ( G ) , since G may be formed by attaching G as a pendant to G at the single vertex v ∈ G .Interchanging the roles of G and G yields µ ( G ) ≤ µ ( G ) as well. (cid:3) Moreover, using similar ideas, we can show that for sufficiently large k , L Nk, ∞ ( G ) is monotoni-cally increasing in k . The na¨ıve intuition behind this is that merging clusters should produce apartition with lower energy, but [BKKM19, Rem. 3.13] shows that things are not that simple.Even the case of p ∈ (0 , ∞ ) requires a relatively fine control of the behaviour of the optimalpartitions, and will be deferred to a later work [HKMP20]. Proposition 4.14.
There exists k ∈ N such that for any k ≥ k ≥ k , (4.7) L Nk , ∞ ( G ) ≥ L Nk , ∞ ( G ) . If G is a tree, then we may take k = 1 .Proof. Let us first give the proof for trees. Fix k ≥ P k = {G , . . . , G k } is an optimal exhaustive k -partition for L Nk, ∞ ( G ), which we know exists by Corollary 4.8. Wewill construct a (test) ( k − P k whose energy is no larger than L Nk, ∞ ( G ), fromwhich we may conclude that L Nk, ∞ ( G ) ≥ L Nk − , ∞ ( G ); the claim of the proposition for trees thenfollows immediately. Suppose without loss of generality that Ω k − and Ω k are neighbours (seeDefinition 2.11). Since G is a tree, they can only meet at a single point, without loss of generalitya vertex v . Moreover, since G k − and G k are connected and G was a tree, the image of v in G i isa single vertex, i = k, k − e G k − byattaching G k to G k − at v , then G k is a pendant of e G k − at v and vice versa, and so, as above,(4.8) µ ( e G k − ) ≤ min { µ ( G k − ) , µ ( G k ) } . Moreover, if we set e Ω k − := Ω k − ∪ Ω k ⊂ G , then e G k − ∈ ρ e Ω k − and the new partition e P := {G , . . . , G k − , e G k − } is a rigid k − G . Combining (4.8) with the definition of Λ N ∞ as a maximum and the fact that no other cluster was affected, we immediately have(4.9) Λ N ∞ ( e P ) ≤ Λ N ∞ ( P k ) . This proves the proposition for trees.The proof for general G is based on the idea above together with the principle that forsufficiently large k , we can always find neighbouring clusters whose supports meet at a singlevertex like Ω k − and Ω k did above. For simplicity, we take k := 4 M , where as usual M is thenumber of edges of G , although this k will in general be far from optimal. Fix k ≥ k + 1and as above denote by P k = {G , . . . , G k } an optimal k -partition for L Nk, ∞ ( G ). Now by the pigeonhole principle, there exists at least one edge of G with non-empty intersection with atleast four cluster supports. It follows that there exist two neighbouring cluster supports, callthem Ω k − and Ω k , which are contained in the interior of this edge. Since their intersectionmust consist of a single point, we may apply verbatim the above argument for trees to thecorresponding clusters G k − and G k to generate a test k − e P with lower energy Λ N ∞ .This completes the proof. (cid:3) Remark 4.15.
As in the Dirichlet case (Remark 4.11), it is easy to construct examples suchas stars for which there is equality in (4.7).5.
Nodal and bipartite minimal Dirichlet partitions
We now wish to consider in detail the relationship between Dirichlet spectral minimal par-titions of a graph G and eigenfunctions of the Laplacian on G , analogous to the results thathave been established in recent years linking partitions of domains Ω with eigenfunctions ofthe Dirichlet Laplacian on Ω, such as discussed in [HHOT09] and related works. On graphs G , however, the correct analogue of the Dirichlet Laplacian on Ω will be the Laplacian withnatural vertex conditions, see Section 2.1.Since in this section we will be working exclusively with the Dirichlet minimisation problemfor rigid partitions, we will not generally distinguish between the cluster supports Ω i ⊂ G andthe clusters G i themselves of a partition P of G , in accordance with Remark 4.4. In such cases,in a slight but simplifying abuse of our own terminology we will speak of the clusters themselvesas being subsets of G . Recall that for the Dirichlet minimisation problem there is no reason toconsider loose partitions.5.1. Nodal and equipartitions.
We begin by introducing two properties of generic loosepartitions of a metric graph.
Definition 5.1.
We say that a k -partition P = {G , . . . , G k } of G is a Dirichlet ( k -)equipartition if λ ( G ) = . . . = λ ( G k ) and a natural ( k -)equipartition if µ ( G ) = . . . = µ ( G k ); or simply an equipartition whenever the spectral problem being considered is clear from the context.If P = {G , . . . , G k } is a Dirichlet equipartition, then its energy Λ Dp ( P ) is independent of p ,being identically equal to λ ( G ); in this case, we will refer to λ ( G ) as the Dirichlet energy ofthe partition, or just energy if the Dirichlet condition is clear from the context. Likewise, if P is a natural equipartition, then µ ( G ) is its natural energy (or just energy ). Definition 5.2.
Let ψ be an eigenfunction associated with µ j ( G ). We call the nodal partitionassociated with ψ the unique internally connected, possibly non-exhaustive partition whosesupport Ω (see Definition 2.9) is the union of edges of G on which ψ does not vanish identically,and whose cut set is the zero set of ψ on the support Ω. We call the cluster supports of thispartition the nodal domains of ψ , and denote by ν ( ψ ) the number of nodal domains. We saythat a given partition is nodal if it is the nodal partition associated with some eigenfunction.Since it is possible for eigenfunctions to vanish identically on one or more edges of a graph,corresponding to a non-exhaustive nodal partition, in such cases there is some freedom as to howexactly to construct a partition out of the eigenfunction; this leads to the following definition.Let us stress once again that λ ( G i ) denotes the lowest eigenvalue of the Laplacian on G i , whereDirichlet conditions are imposed at all cut points. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 29
Definition 5.3.
Let P = {G , . . . , G k } be a k -partition of G . Then we say that P is a generalisednodal partition if there exist eigenfunctions ψ , . . . , ψ k for λ ( G ) , . . . , λ ( G k ) with the followingproperties:(1) for each i , there exists a cut e G i of G i such that on each connected component of e G i either ψ i is identically zero, or the connected component is the closure of a nodal domain of ψ i on G i ; and(2) the k -partition e P of G , k ≥ k , consisting of all connected components of e G i on which ψ i is not identically zero, for all i = 1 , . . . , k , is a nodal partition associated with someeigenfunction of G .At the risk of being redundant, let us elaborate on Definition 5.3. As P may fail to beinternally connected, Dirichlet conditions may be imposed on vertices of clusters G i in sucha way that G i is de facto disconnected; and in particular, the lowest eigenvalue need not besimple and the ground state need not be strictly positive; indeed it may vanish identically onan edge, as the following examples show (see also Example 4.10). Example 5.4. (1) Let G be the equilateral pumpkin on 3 edges of length 1 (Example 4.1);then the eigenvalue µ ( G ) = π has multiplicity three. If ψ is taken as the eigenfunction whichis monotonic on each edge and invariant under permutations of the edges ( longitudinal , in thelanguage of [BKKM19, Section 5.1]), then the corresponding nodal partition is an exhaustivefaithful 2-partition of G whose clusters are both 3-stars with edges of length 1 / ψ isa transversal eigenfunction, supported on two of the edges and identically zero on the third,then the nodal partition is a non-exhaustive rigid 2-partition of graph, whose clusters are eachedges of G (and the third edge is not in the cluster support). If we take ψ a linear combinationof transversal eigenfunctions which has its zeros at the vertices of G , is positive on two edgesand negative on the third, then the result is an exhaustive rigid 3-partition whose clusters arethe edges of G .(2) If we now take G to be a star on 3 edges e , e , e of lengths 1, 1 and ε ∈ (0 , µ ( G ) = π has multiplicity one, with eigenfunction supported on e ∪ e . The uniquecorresponding nodal partition has clusters e and e ; e is not in the support of the eigenfunctionand hence not in the support of the partition. However, the exhaustive 2-partition P = { e , e ∪ e } , which is rigid but not internally connected, is a generalised nodal partition,as we can recover the nodal partition upon removing the extraneous edge e on which theeigenfunction vanishes from the cluster e ∪ e .Thus a partition P is a generalised nodal partition of G if there exists an eigenfunction whoseeigenvalue equals the energy of the partition, and whose nodal domains correspond exactly tosubsets of clusters of P – but there may be parts of these clusters on which the eigenfunctionvanishes identically. The non-exhaustive partition obtained by removing the latter parts is thennodal. We will return to a slightly different aspect of extracting such nodal-type partitions frommore general partitions in Proposition 8.9.All nodal partitions are rigid, since no eigenfunction on G has an isolated zero withoutchanging sign in a neighbourhood of it; this fact follows from the fact that the eigenfunctionsatisfies the Kirchhoff condition at every point of the graph (see also [Kur19] for a more generaldiscussion of eigenfunction positivity). Furthermore, all nodal partitions are Dirichlet equipar-titions whose energy is the associated eigenvalue. In fact, more generally, the Dirichlet energyΛ D ∞ ( P ) of any generalised nodal partition is necessarily equal to the corresponding eigenvalue.Next, we extend to metric graphs two relationships between Laplacian eigenvalues and optimal Dirichlet partitions which are well known in the case of domains (see [BNH17, Proposition 10.6and eq. (10.44)]), and give a partial extension to metric graphs of Courant’s Nodal DomainTheorem.
Proposition 5.5.
We have µ k ( G ) ≤ L Dk, ∞ ( G ) .Proof. Let us denote by P = {G , . . . , G k } an arbitrary loose k -partition and by ϕ , . . . , ϕ k the normalised positive ground states of G , . . . , G k , respectively; that is, ϕ i is the positiveeigenfunction associated with λ ( G i ), i = 1 , . . . , k , chosen to have L -norm 1. Let us denoteby ψ , . . . , ψ k − orthonormalised eigenfunctions associated with µ ( G ) , . . . , µ k − ( G ) respectively.We set, for a fixed k -tuple ( t , . . . , t k ) ∈ R k which will be specified later, φ := t ϕ + · · · + t k ϕ k . The system of equations h ψ i , φ i = k X j =1 t j h ψ i , ϕ j i = 0has size ( k − × k and so rank at most k −
1. Hence there exists ( t , . . . , t k ) ∈ R k such that h ψ i , φ i = 0 for all i ∈ { , . . . , k − } and k P j =1 t j = 1. Then, from the variational characterisationof the eigenvalues, µ k ( G ) ≤ ˆ G ( φ ′ ) d x = k X j =1 t j ˆ G j ( ϕ ′ j ) d x = k X j =1 t j λ ( G j ) ≤ Λ D ∞ ( P ) ≤ L Dk, ∞ ( G ) . This concludes the proof. (cid:3)
Proposition 5.6.
We have µ ( G ) = L D , ∞ ( G ) , and any partition realising L D , ∞ ( G ) is a gener-alised nodal partition.Proof. Let ψ be an eigenfunction associated with µ ( G ), P its associated nodal partition and ν the cardinality of P . The function ψ is orthogonal to the constants in L ( G ) and thereforechanges sign. It follows that ν ≥ L D , ∞ ( G ) ≤ L Dν, ∞ ( G ) = µ ( G ) by Remark 4.11.(1),while µ ( G ) ≤ L D , ∞ ( G ) according to Proposition 5.5. We have shown that L D , ∞ ( G ) = µ ( G ).Let us now consider a partition P ∗ = {G ∗ , G ∗ } realising L D , ∞ ( G ) and let us denote by ϕ ∗ and ϕ ∗ the normalised positive ground states of G ∗ and G ∗ . There exists ( t , t ) ∈ R such that ψ = t ϕ ∗ + t ϕ ∗ is orthogonal to the constants and t + t = 1. Then, from the variationalcharacterisation of the eigenvalues, µ ( G ) ≤ ˆ G ( ψ ′ ) d x = t ˆ G ∗ (( ϕ ∗ ) ′ ) d x + t ˆ G ∗ (( ϕ ∗ ) ′ ) d x = t µ ( G ∗ ) + t µ ( G ∗ ) ≤ L D , ∞ ( G ) . Since L D , ∞ ( G ) = µ ( G ), the above inequality implies that P ∗ is an equipartition and µ ( G ) = ´ G ( ψ ′ ) d x . By the variational characterisation, ψ is an eigenfunction associated with µ ( G ). (cid:3) Proposition 5.7 (Weak Courant Theorem) . Given an eigenvalue µ k ( G ) and an associatedeigenfunction ψ , denote by κ ( µ k ( G )) the integer κ ( µ k ( G )) := max { j ∈ N : µ j ( G ) = µ k ( G ) } , and by ν ( ψ ) the number of nodal domains of ψ . Then ν ( ψ ) ≤ κ ( µ k ( G )) . THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 31
This theorem is well known for domains, where in fact the stronger statement ν ( ψ ) ≤ k alwaysholds, and has been the subject of much work since Pleijel’s groundbreaking paper [Ple56];see, for example, [BNH17, Section 10.2.4] or [L´en19] and the references therein. This weakerversion also holds for discrete graph Laplacians; see [DGLS01, Theorem 2]. For metric graphsfor which all Laplacian eigenvalues are simple, κ ( µ k ( G )) = k , while ν ( ψ ) is the correspondingnodal count.Hence the above inequality reduces to the main result in [GSW04], obtained formore general Schr¨odinger operators. Proof.
To simplify notation, we set ν := ν ( u ) and κ := κ ( µ k ( G )). Let us assume for a con-tradiction that ν ≥ κ + 1. We denote by P the nodal partition associated with u . We have µ κ ( G ) = Λ D ∞ ( P ) ≥ L Dν, ∞ ( G ). According to Proposition 5.5, µ ν ( G ) ≤ L Dν, ∞ ( G ). Since µ κ +1 ( G ) ≤ µ ν ( G ) and µ κ ( G ) < µ κ +1 ( G ), by definition of κ = κ ( µ k ( G )), we obtain µ κ ( G ) < µ κ ( G ). (cid:3) Bipartite minimal partitions.
As observed in [BBRS12], the links between minimaland nodal partitions appear more clearly if we restrict ourselves to partitions that are proper.The following theorem can be deduced immediately from results in [BBRS12]. We state it usingour notation for future reference.
Definition 5.8.
Let P be a partition of G . We say that P is bipartite if each of its clusters canbe marked with signs + or − in such a way that any two clusters have different signs if theirsupports are neighbours. Theorem 5.9.
Let P be an exhaustive, proper Dirichlet minimal k -partition of G , that is, suchthat Λ D ∞ ( P ) = L Dk, ∞ ( G ) . Then P is bipartite if and only if it is nodal. Note that the assumptions that P is proper and Dirichlet minimal imply that P is necessarilyexhaustive. Proof. If P is nodal, we see that it is bipartite by using the sign of the corresponding eigen-function and the assumption that P is proper. Conversely, let us assume that P is bipartite.Since P is minimal, it is in particular critical for the functional Λ introduced in Definition 2.6in [BBRS12] and corresponding to Λ D ∞ in our notation. From [BBRS12, Theorem 2.10(1)], weconclude that P is nodal, under the condition that k is large enough. According to [BBRS12,Theorem 5.2], the conclusion actually holds without this last condition. (cid:3) The assumption that P is proper is crucial in the previous theorem. However, if G is a tree ,then any (exhaustive) equipartition is nodal, since we can recursively construct an eigenfunctionout of the ground states of the clusters. Theorem 5.10.
Let G be a compact tree and suppose P = {G , . . . , G k } is an exhaustiveDirichlet k -equipartition of G , k ≥ . Then P is nodal. We note explicitly that P does not have to be a minimal partition of G for the theorem tohold: important is merely the equipartition property.While Theorem 5.10 is obvious for proper partitions (and indeed the corresponding eigen-function is the k -th eigenfunction of G , see [Ban14]), the proof in the general case requires morework. We first wish to establish some basic structural properties of partitions of trees, sincethe proof will consist of gluing together the eigenfunctions of the individual clusters recursivelyin the right way. We will use the terminology introduced in Section 2.3, in particular Defini-tions 2.9, 2.10 and 2.11, without further comment; in particular, we assume without loss ofgenerality that the cut set of a partition consists only of vertices of G . Lemma 5.11.
Under the assumptions of Theorem 5.10 the following assertions hold.(1) Each G i is itself a tree, i = 1 , . . . , k .(2) Any two cluster supports share at most one separating point, that is, for all i = j , Ω i ∩ Ω j consists of at most one vertex.(3) There exists at least one i = 1 , . . . , k for which ∂ Ω i consists of exactly one vertex.Proof. (1) Trivial.(2) Follows from a simple argument tracing paths between clusters.(3) Suppose for a contradiction that for each i = 1 , . . . , k , ∂ Ω i contains at least two vertices.We will construct a loop in G using (2). Start at any cut point, call it v , choose a neighbouringsupport Ω and a second vertex v = v ∈ ∂ Ω . Let γ be (the unique image in G of) acontinuous, injective mapping of [0 ,
1] into Ω which goes from v at 0 to v at 1. Now, at v ,pick some other neighbour Ω = Ω with another vertex v = v ∈ ∂ Ω and an injective path γ from v to v within Ω . Repeat this process inductively: then, for some 2 ≤ m ≤ k , wemust have that Ω m = Ω i for some 1 ≤ i ≤ m . At this point, on choosing v m = v i − , we seethat γ i ∪ . . . γ m forms a closed path from v i − to itself, which is injective except at at most afinite number of vertices. This contradicts the assumption that G was a tree. (cid:3) We may imagine any cluster support satisfying condition (3) to be at the “end” of the graphin some sense, and build a natural hierarchy of neighbours emanating from it. More precisely,suppose Ω is any such support. We will refer to the level of any support within the samepartition, with respect to Ω , via the rules: • Ω has level zero; • any neighbour of Ω has level one; • the level of any support other than Ω is the minimum of the levels of its neighbours,plus one.Thus the level is, loosely speaking, the number of supports we need to traverse to reach Ω (excluding Ω itself). By Lemma 5.11, this is well defined, and indeed any support Ω i m of level m ≥ m − m and/or m + 1 as neighbours. It always sharesa separating point v m with exactly one support Ω i m − of level m − ∂ Ω i m ∩ ∂ Ω i m − = { v m } ;moreover, if Ω j m is any other neighbour of Ω i m of level m , then also ∂ Ω i m ∩ ∂ Ω j m = { v m } .With this background and terminology, we can give the proof of Theorem 5.10. Proof of Theorem 5.10.
Set µ := λ ( G ) = . . . = λ ( G k ), denote by Ω , . . . , Ω k ⊂ G the clustersupports corresponding to G , . . . , G k , and denote by ψ , . . . , ψ k ∈ H ( G ) the functions sup-ported on Ω i , i = 1 , . . . , k , which correspond to the eigenfunctions of G , . . . , G k , respectively,normalised to have L -norms one; that is, ψ i satisfies − ψ ′′ ( x ) = µψ ( x ) in the interior of eachedge of Ω i , and ψ ( x ) = 0 for all x ∈ G \ Ω i .We will define an eigenfunction ψ on G of the form(5.1) ψ ( x ) = k X i =1 t i ψ i ( x )for all x ∈ G , for coefficients t i ∈ R \ { } to be chosen in such a way that ψ satisfies theKirchhoff condition at each vertex of G . (Since the Ω i are pairwise disjoint, it is clear from theoutset that − ψ ′′ = µψ on each edge and ψ satisfies the continuity condition at each vertex of G ; hence, to check that ψ is an eigenfunction on G for µ , we only need to check the Kirchhoffcondition.) THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 33
To this end, we first note that at every boundary vertex v ∈ V D , for any edge e incident to v , if, say, e ⊂ Ω i , then by Lemma 5.11 and the fact that ψ i is strictly positive everywhere inΩ i (see [Kur19, Theorem 3]), the outer normal derivative ∂ ν ψ i | e ( v ) of ψ i on e at v is differentfrom zero.Based on this observation and the level structure of the supports Ω i as established inLemma 5.11, a routine induction on the level can be used to specify the t i . (cid:3) We finish this subsection with an example of an exhaustive minimal partition which is not anequipartition, nor is it bipartite; but it still corresponds to a nodal partition. This illustratesthe difficulties in extending Theorem 5.9 to non-proper partitions; in particular, there does notseem to be a natural concept of “bipartite” partitions that would allow the theorem to hold.
Example 5.12.
Suppose that H ε is the 3-star (cf. Example 4.10) having edges e , e and e of length 1 + ε , 1 and 1, respectively, for some small ε ≥
0, joined at a common vertex v .Then the unique internally connected partition achieving L D , ∞ ( H ε ) has { v } as its cut set, with G i = e i , i = 1 , , π /
4; this corresponds to the eigenvalue µ ( H ε ), whoseeigenfunction is supported on the edges of length 1 and identically zero on e : in particular,this eigenfunction has a nodal pattern corresponding to the partition. Note that this is not anequipartition if ε >
0, in opposition to domains where a Dirichlet minimal partition is always anequipartition (see [BNH17, Proposition 10.45]). If ε = 0, then this is a (Dirichlet) equipartitionand corresponds to the nodal partition for µ ( H ) = µ ( H ) (more precisely, it is nodal for µ ( H ), and generalised nodal with respect to µ ( H )). However, it is not bipartite.5.3. Courant-sharp eigenfunctions.
The next theorem, like Theorem 5.9, was proved in[BBRS12], but here we wish to give a fundamentally different proof, based on a continuityargument involving the introduction of Robin-type conditions at the cut set of the partition;we imagine that such ideas might also be adaptable to the corresponding problem on domains.
Theorem 5.13.
Given a positive integer k , suppose there exists a k -partition of G realising L Dk, ∞ ( G ) which is proper and nodal. Then L Dk, ∞ ( G ) = µ k ( G ) .Proof. Let P be a nodal, minimal and proper (thus necessarily exhaustive) k -partition, and letus denote the corresponding eigenvalue by µ , so that µ = L Dk, ∞ ( G ). Let ψ be an eigenfunctionassociated with µ . We now define a family of vertex conditions in the following way; we willdenote the corresponding (quantum) graphs by ( G θ ) θ ∈ [0 ,π/ : at each cut point v of the partition P , which by assumption is a vertex of degree two, we impose the Robin-type vertex condition(5.2) cos θ ( f ′ ( v − ) + f ′ ( v + )) = (sin θ ) f ( v ) , with f additionally required to be continuous. (In fact, the condition (5.2) corresponds toputting a δ -potential of strength tan θ at v , for θ ∈ [0 , π/ θ = π/ j ≥
1, we write µ j ( θ ) := µ j ( G θ ). Note that when θ = 0, (5.2)reduces to the usual Kirchhoff condition, so that µ j (0) = µ j ( G ) for all j ≥
1. We then have thefollowing further basic properties.
Lemma 5.14.
The family of graphs ( G θ ) θ ∈ [0 ,π/ has the following properties.(1) For all integers j ≥ , the function θ µ j ( θ ) is continuous and non-decreasing on [0 , π/ .(2) For all integers ≤ j ≤ k , µ j ( π/
2) = µ , and µ k +1 ( π/ > µ .(3) For all θ ∈ [0 , π/ , µ is an eigenvalue of the graph G θ , with ψ an associated eigenfunc-tion. We will not give a detailed proof: for part (1), cf. [BK13, Section 3.17] or [BKKM19, The-orem 3.4]; for (2), note that if we treat Dirichlet points as cut points, then G π/ has exactly k connected components, corresponding to the clusters of P , each having the same eigenvalue µ with multiplicity one; for (3), we simply note that each v is a zero of ψ , so that the vertexcondition (5.2) reduces to the usual Kirchhoff condition.We denote by U the subspace of the eigenspace of G = G associated with µ , whose functionsvanish at each cut point of P . We have ψ ∈ U , but U could contain functions which are notproportional to ψ , in particular if µ has multiplicity greater than one. For all θ ∈ [0 , π/ U is contained in the eigenspace of the graph G θ associated with the eigenvalue µ . Conversely, if ϕ is an eigenfunction of the graph G θ for some θ ∈ [0 , π/
2] and if ϕ vanishes at each cut pointof P , then ϕ ∈ U . This means that U can alternatively be described as the intersection of allthe eigenspaces of G θ , associated with µ , for all θ ∈ [0 , π/ ℓ the smallest positive integer such that µ = µ ℓ ( G ); it follows from Propo-sition 5.5 and the assumption µ = L Dk, ∞ ( G ) that ℓ ≥ k . Let us assume for a contradiction that ℓ > k , which is equivalent to µ k (0) < µ . Lemma 5.15.
Under this assumption, there exists a smallest ¯ θ := θ ∈ (0 , π/ such that µ k ( θ ) = µ , and an eigenfunction ϕ of G ¯ θ associated with µ , which does not belong to U .Proof of Lemma 5.15. According to property (1) of Lemma 5.14, µ k ( θ ) → µ k ( π/
2) = µ and µ k +1 ( θ ) → µ k +1 ( π/
2) as θ → π/
2. Suppose that µ k ( θ ) < µ for all θ ∈ (0 , π/ µ is also an eigenvalue for each θ by (2), that is, for each θ there exists some ℓ ( θ ), necessarilyno smaller than k + 1, such that µ = µ ℓ ( θ ) ( θ ), by continuity we would also have µ k +1 ( θ ) ≤ µ ,and thus, from (1), µ k +1 ( π/ ≤ µ . This contradicts (2), and we conclude that there exists θ ∈ (0 , π/
2) such that µ k ( θ ) = µ . We can then choose ¯ θ to be the greatest lower bound of allsuch θ .For each θ ∈ [0 , ¯ θ ), we pick an eigenfunction ϕ θ of G θ associated with µ k ( θ ), normalised tohave L -norm one. We claim that ( ϕ θ ) θ admits a convergent subsequence as θ → ¯ θ . Indeed,since each function satisfies − ϕ ′′ θ = µ k ( θ ) ϕ θ edgewise, and ( µ k ( θ )) θ is bounded in R and ( ϕ θ ) θ is bounded in L ( G ), there exists a subsequence converging weakly in H and hence stronglyin H – in fact in C – on each edge to a function ϕ . From the Rayleigh quotients, possiblyup to a further subsequence (which we will still simply denote by ϕ θ ) we also see that theconvergence is weak in H ( G ). In particular, by compactness of the injection H ( G ) ֒ → C ( G ),the limit function ϕ is continuous. The strong C -convergence on each edge implies that ϕ alsosatisfies the condition (5.2) at each vertex for θ = ¯ θ . From the fact that ´ G | ϕ ′ | d x ´ G | ϕ | d x = lim θ → ¯ θ µ k ( θ )and an inductive argument involving convergence of the eigenfunctions of the lower eigenfunc-tions, we can finally conclude that ϕ is in fact an eigenfunction of G ¯ θ associated with µ k (¯ θ ),and in particular also with µ = µ k (¯ θ ).Finally, for each θ < ¯ θ , since ϕ θ is associated with µ k ( θ ) < µ and U is contained in theeigenspace associated with the eigenvalue µ , we have that ϕ θ U , and in fact ϕ θ is orthogonalto each function of U . Strong convergence ϕ θ → ϕ in L ( G ) now implies that ϕ is orthogonalto every function in U . (cid:3) An example for such a G is the equilateral 3-star with edges of length 1 each, where we take µ = π / µ ( G ) = µ ( G ) and we consider any partition whose cut set is the central vertex of G . THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 35
Now, since ϕ / ∈ U , there exists at least one cut point of P where ϕ does not vanish, which wedenote by v . For ε > ψ ε := ψ + εϕ has exactly one zero close toeach cut point of P , and does not vanish elsewhere. Therefore, the nodal partition associatedwith ψ ε , which we denote by P ε , is a proper k -partition. Let us denote the clusters of P ε by G ( ε ) i , with i ∈ { , . . . , k } , and the restriction of ψ ε to G ( ε ) i by ψ ( ε ) i . There is one cluster G ( ε ) i whose interior contains the cut point v .Let us now consider a general cluster G ( ε ) i and denote by { v , . . . , v ℓ } the cut points containedin its interior (it is possible that there is no such cut point, in which case ℓ = 0 and this set isempty). We have ˆ G ( ε ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ψ ( ε ) i (cid:17) ′ (cid:12)(cid:12)(cid:12)(cid:12) dx = µ ˆ G ( ε ) i (cid:12)(cid:12)(cid:12) ψ ( ε ) i (cid:12)(cid:12)(cid:12) dx − tan θ ℓ X j =1 (cid:16) ψ ( ε ) i ( v j ) (cid:17) = µ ˆ G ( ε ) i (cid:12)(cid:12)(cid:12) ψ ( ε ) i (cid:12)(cid:12)(cid:12) dx − ε tan θ ℓ X j =1 ϕ ( v j ) ≤ µ ˆ G ( ε ) i (cid:12)(cid:12)(cid:12) ψ ( ε ) i (cid:12)(cid:12)(cid:12) dx. By the variational characterisation of the eigenvalues, λ ( G ( ε ) i ) ≤ µ . In the particular case ofthe cluster G ( ε ) i , the sum on the right-hand side is strictly positive, and therefore λ ( G ( ε ) i ) < µ .We obtain Λ D ∞ ( P ε ) ≤ Λ D ∞ ( P ), with λ ( G ( ε ) i ) < Λ D ∞ ( P ). We have reached a contradiction: either λ ( G ( ε ) i ) < Λ D ∞ ( P ) for all 1 ≤ i ≤ k , in which case P is not minimal, or λ ( G ( ε ) i ) = Λ D ∞ ( P )for some 1 ≤ i ≤ k , in which case P ε is minimal without being an equipartition, contradictingTheorem 5.9. (cid:3) Minimal partitions for non-Courant-sharp eigenfunctions.
We now consider anexample showing that for k ≥
3, a nodal partition which achieves L Dk, ∞ ( G ) is not necessarilygiven by an eigenfunction associated with µ k ( G ). This is in contrast to the situation for domainsin R (see [HHOT09, Theorem 1.17]).We will consider a pumpkin graph on three edges, of length π , 2 π and 2 π , respectively; inother words, we consider a graph H with two vertices { v, w } and three edges { e , e , e } . Weset e = [ x , x + π ], e = [ x , x + 2 π ] and e = [ x , x + 2 π ] (see Figure 5.1). v we e e Figure 5.1.
The pumpkin graph H after insertion of two dummy vertices (topand bottom, corresponding to the points x + π and x + π , respectively) tomake it equilateral (left); an optimal 4-partition of H , where the thick black dotsdenote the partition cut set, which divides D into two equilateral 3-stars whoseedges all have length π/
2, and two intervals of length π each (right).Because the edges of H have rationally dependent lengths, we can easily exploit von Below’sformula to compute the eigenvalues of the Laplacian on H . More precisely, insertion of two dummy vertices turns H into an equilateral metric graph on four vertices and five edges, eachof length π . The underlying discrete graph has transition matrix T = − − − − − − − − − − whose eigenvalues are µ j = 1 , , − , − . In view of [Bel85, Theorem in §
5] and by rescaling,the eigenvalues of H are • D is connected); • the infinitely many values attained by (cid:0) π arccos µ j (cid:1) , µ j = 1, from the three non-trivialeigenvalues of T ; • k with multiplicity 3 for even k ; • k with multiplicity 1 for odd k .In particular, the fifth-lowest eigenvalue of the Laplacian on H with natural vertex conditionsis 1 and is simple. Moreover, one can check directly that its eigenfunction vanishes identicallyon the edge e , and at the four vertices marked in Figure 5.1-left, for a total of four nodaldomains. The following lemma summarises these statements. Lemma 5.16.
The eigenvalue µ ( H ) is simple and equal to . In addition, if ψ is an associ-ated eigenfunction, the nodal (non-exhaustive) -partition associated with ψ is, in the notationintroduced above (see Figure 5.1): N := { [ x , x + π ] , [ x + π, x + 2 π ] , [ x , x + π ] , [ x + π, x + 2 π ] } . Proposition 5.17.
We have L D , ∞ ( H ) = µ ( H ) = 1 . An optimal (exhaustive, internally connected, equilateral) partition corresponding to L D , ∞ ( H )is depicted in Figure 5.1 (right). Note that this also equals L D , ∞ ( H ), which is realised by thecut set indicated in Figure 5.1 (left). Our proof will show that any minimal partition must haveenergy 1. In fact, with a little more effort, one could show that even when minimising amongall not necessarily exhaustive partitions, the minimal energy is still 1, even though the set ofminimal partitions is larger (as Figure 5.1 demonstrates). In Section 8.2 we compare exhaustiveand non-exhaustive Dirichlet minimal partitions, and refer in particular to Conjecture 8.7. Proof of Proposition 5.17.
Let P = {G , G , G , G } be a partition achieving L D , ∞ ( H ). It followsfrom Lemma 5.16 and Remark 4.11 that Λ D ∞ ( P ) = L D , ∞ ( H ) ≤
1. In the rest of the proof, for i ∈ { , , , } and j ∈ { , , } , we say that the cluster G i is contained (resp. strictly contained )in e j when G i ⊂ e j (resp. G i ( e j ), and we say that G i meets e j when int G i ∩ int e j = ∅ . (Herewe emphasise that we are identifying G i with Ω i ⊂ G .) Since the energy of P is at most 1, wesee easily that e cannot strictly contain a cluster of P ; in particular any cluster contained in e has to be equal to e . Similarly, if e contains two clusters, they are both intervals of length π and their union is equal to e . The same holds for e .Let us now discuss the possible cases. Since our partition is by assumption exhaustive, atleast one cluster of P meets e . Up to relabelling of the supports and without loss of generality,we can assume that G meets e . Let us first consider the case where G is contained in e .Then G = e , and { v, w } is contained in the cut set of P . Each of the other clusters is thuscontained either in e or in e . From the preliminary remarks, there is, up to relabelling of THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 37 the edges and the support, only one possiblity: e is the union of G and G , which are bothintervals of length π , and G = e . This gives energy exactly 1.Up to relabelling of the clusters, there thus remains only one case: G meets e without beingcontained in e . Then one of the vertices { v, w } is contained in int G . Without loss of generality,we can assume v ∈ int G . If no other cluster meets e , then either e or e must strictly containtwo clusters, in contradiction to the preliminary remarks. Without loss of generality we canassume that G meets e , and therefore that w ∈ int G . Since P is rigid, it follows that G and G are intervals of length at least π contained in int e and int e , respectively, and {G , G } is a partition of H \ G ∪ G , whose cut set consists of a single point in e . We construct anew graph e H from H \ G ∪ G by gluing together the two degree one vertices correspondingto the two extremities of G , and likewise gluing together the two vertices corresponding tothe extremities of G . To be more explicit, let us set G = [ y , z ] and G = [ y , z ], with x < y < y + π ≤ z < x + 2 π and x < y < y + π ≤ z < x + 2 π . The graph e H thenhas three edges e e := e , e e = [ x , x + 2 π − z + y ] and e e = [ x , x + 2 π − z + y ]; andtwo vertices, e v := v and e w = { x + π, x + 2 π − z + y , x + 2 π − z + y , π − z + y } . Itis a 3-pumpkin graph with edges of length at most π each and e P := {G , G } is a (internallyconnected) 2-partition of e H . From Proposition 5.6, µ ( e H ) = L D , ∞ ( H ) ≤ Λ D ∞ ( e P ) ≤ . By monotonicity of the eigenvalues with respect to edge length, see [BKKM19, Corollary 3.12(1)],we have, for any integer j ≥
1, 1 = µ j ( H ∗ ) ≤ µ j ( e H ), where H ∗ is the 3-pumpkin graph withedges of length π . In particular, L D , ∞ ( e H ) = µ ( e H ) ≥
1. From this we conclude that Λ D ∞ ( P ) = 1in the case where v ∈ int G and w ∈ int G . We have seen that, in all cases, we necessarilyhave Λ D ∞ ( P ) = 1. (cid:3) Non-bipartite minimal partitions.
The goal of this section is to show that any properminimal partition of a metric graph is the projection of a nodal partition on a double covering.To reach it, we have to give some additional definitions. We first recall the (standard) definition:given a metric graph G (the base graph ), a double covering is another metric graph b G (the covering graph ) equipped with a surjective map Π : b G → G (the covering map ). We requirethat(1) Π is locally an isometry;(2) for all x ∈ G , Π − ( { x } ) contains two elements.Furthermore, given such a covering, we define the deck map σ : b G → b G by σ ( y ) = y andΠ( σ ( y )) = Π( y ) for all y ∈ b G . The map σ clearly satisfy σ ◦ σ = Id and is locally an isometry,and therefore is globally an isometry of b G .We have the orthogonal decomposition L ( b G ) = S ( b G ) ⊕ A ( b G ) , where S ( b G ) and A ( b G ) are the subspaces of functions f ∈ L ( b G ) satisfying respectively f ◦ σ = f or f ◦ σ = − f . This decomposition is preserved by b L , the natural Laplacian on b G , in the followingsense. For any sufficiently regular function f , b L ( f ◦ σ ) = b L ( f ) ◦ σ . It follows that the domain b D of b L also has an orthogonal decomposition: b D = ( b D ∩ S ( b G )) ⊕ ( b D ∩ A ( b G )) . Accordingly, we define the operator b L a as the restrictions of b L to b D ∩ A ( b G ). It is self-adjointwith compact resolvent in the Hilbert space A ( b G ). Its spectrum therefore consists of a sequenceof eigenvalues with finite multiplicity which we denote by ( µ aj ( b G )) j ≥ (counting multiplicities).Let us now consider a partition P of G . For each cluster G i , Π − ( G i ) is a closed set having atmost two connected components. The collection of all these connected components when i runsover all the possible values is a partition of b G , once we have given it an (arbitrary) indexation.We denote it by b P and call it the partition of b G lifted from P . Theorem 5.18.
Let P be a proper Dirichlet minimal non-bipartite k -partition of G .(1) There exists a double covering b G such that the lifted partition b P is the nodal partitionof an eigenfunction of b L a .(2) For any such double covering, Λ D ∞ ( P ) = µ ak ( b G ) .Proof. Most of the work goes into proving (1). We follow closely the argument in the proof of[BBRS12, Theorem 2.10(1)] (see Section 4.1 of that reference). Since P is Dirichlet minimal,it is a critical point for the functional Λ, with the parametrisation described by [BBRS12,Theorem 2.8]. More precisely, this means that the following holds, according to [BBRS12,Section 4.1].We choose a set of edges { e j ; 1 ≤ j ≤ β } whose removal turns G into a tree. We choose(arbitrarily) a point v j in the interior of each e j . These are called section points in [BBRS12].We split each v j into two vertices v − j and v + j according to the orientation of e j , v − j and v + j being the end of the left and right part of e j respectively. We obtain a metric tree T . For any( ϕ , . . . , ϕ β ) ∈ ( − π, π ] β , we define the self-adjoint operator on T acting as the (opposite of) thesecond derivative, and whose domain consistes of the functions f ∈ ⊕ e ∈E ( T ) H ( e ) which satisfythe standard boundary conditions at the vertices of G and, for all j ∈ { , . . . , β } , ( cos ( ϕ j / f ′ ( v − j ) = − sin ( ϕ j / f ( v − j ) , cos ( ϕ j / f ′ ( v + j ) = sin ( ϕ j / f ( v + j ) . In this way, we have defined a quantum graph (i.e. triple of metric graph, differential expressionand vertex conditions) that we denote by T ( ϕ ,...,ϕ β ) . According to [BBRS12, Theorem 2.8],there exists ( ϕ , . . . , ϕ β ) ∈ ( − π, π ] β such that P is the nodal partition of an eigenfunction ψ of T ( ϕ ,...,ϕ β ) associated with a simple eigenvalue. Furthermore, according to [BBRS12, Section4.1], | ψ ( v + j ) | = | ψ ( v − j ) | and | ψ ′ ( v + j ) | = | ψ ′ ( v − j ) | for all j ∈ { , . . . , β } .For each j ∈ { , . . . , β } , we denote by C j the shortest path in the metric graph T whichconnects the vertices v − j and v + j . We have the following alternative.(A) ψ ( v ± j ) = 0 and C j contains an even number of zeros of ψ or ψ ( v ± j ) = 0 and C j containsan odd number of zeros of ψ . In that case, ψ ( v + j ) = ψ ( v − j ) and ψ ′ ( v + j ) = − ψ ′ ( v − j ).(B) ψ ( v ± j ) = 0 and C j contains an odd number of zeros of ψ or ψ ( v ± j ) = 0 and C j contains aneven number of zeros of ψ . In that case, ψ ( v + j ) = − ψ ( v − j ) and ψ ′ ( v + j ) = ψ ′ ( v − j ).We construct the double cover b G by gluing two copies of e G , denoted by T u and T d , accordingto the following rules. For each j ∈ { , . . . , β } , we denote by v − j,u , v + j,u , v − j,d and v + j,d the verticescorresponding to v − j and v + j in T u and T d respectively.(1) In Case (A), we identify back v − with v +1 and v − with v +2 , that is we glue back the cutpoints in each copy separately. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 39 (2) In Case (B), we identify v − with v +2 and v − with v +1 .The covering map Π : b G → G is defined as the unique continuous extension of the map sendingeach point in T u or T d , distinct from the selected section points, to the corresponding point in G . This map is a local isometry and each of its fibres has two elements.We then define the function b ψ as the continuous extension to b G of the function which isequal to ψ on T u and − ψ on T d (this continuous extension exists by construction of b G ). Byconstruction, b ψ is an antisymmetric eigenfunction of standard Laplacian on b G , associated withthe eigenvalue Λ D ∞ ( P ). Furthermore, its nodal partition is b P . This conclude the proof of Part(1).In order to prove Part (2), we repeat the proof of Theorem 5.13, replacing the Hilbertspace L ( G ) with A ( b G ), and more generally all the function spaces on G by the correspondingantisymmetric function spaces on b G . (cid:3) For readers familiar with magnetic Schr¨odinger operators on metric graphs, we mention thatthe previous construction has a magnetic interpretation. Indeed, b L a is unitarily equivalent toa magnetic Laplacian, by which we mean a magnetic Schr¨odinger operator with zero potential(see [BK13, Section 2.6] for the relevant definitions). According to [BK13, Corollary 2.6.3], suchan operator is defined, up to unitary equivalence, by specifying the magnetic flux, modulo 2 π ,through each cycle of G . For this, we use the following rule. We count the number of times thecycle crosses a cut-point of P . The flux is π if this number is odd and 0 if it is even. Althoughwe will not go further into details, let us also mention that for such a magnetic Laplacian, P isthe nodal partition of an eigenfunction associated with the k -th eigenvalue.5.6. Comparison with two-dimensional domains.
In order to put the results of this sec-tion into perspective, let us compare them with those previously obtained for minimal partitionsof two-dimensional domains. We first briefly review this last theory, as developed in [HHOT09].The interested reader can find a more extensive survey (including numerical results) and a muchmore complete bibliography in [BNH17].Following [HHOT09], we take Ω ⊂ R to be open, bounded and connected, with a piecewise C ,α boundary for some α > domain . In this subsection only, we calla k -partition of Ω a family P = { Ω , . . . , Ω k } of k subsets of Ω which are mutually disjoint,connected and open. Comparing this definition with Section 2, we see that partitions aredefined in [HHOT09] by their cluster supports, taken to be open sets. The following exampleis particularly important. Let ψ be an eigenfunction of − ∆ Ω , the Dirichlet realisation of theLaplacian in Ω. The nodal partition associated with ψ is the family of all the nodal domainsof ψ , that is the connected components of the complement of its zero set. This is analogous toDefinition 5.2.We denote by P k (Ω) the set of all k -partitions of Ω in the above sense. The followingdefinitions are in complete analogy with Section 4. We first set, for a given partition P ,(5.3) Λ Dp ( P ) := (cid:18) k k P i =1 λ (Ω i ) p (cid:19) /p if p ∈ (0 , ∞ ) , max i =1 ,...,k λ (Ω i ) if p = ∞ , where λ (Ω i ) denotes the first eigenvalue of − ∆ Ω i . We then define(5.4) L Dk,p (Ω) := inf P∈ P k (Ω) Λ Dp ( P ) and call minimal any k -partition P satisfying Λ Dp ( P ) = L Dk,p (Ω). It was proved in [HHOT09],building on results from [BBH98, CTV05, CL07], that minimal partitions exist for any positiveinteger k and any p ∈ [1 , ∞ ], and are very regular. In the terminology of [HHOT09], they are strong , meaning that Ω = k [ i =1 Ω i ;in our terminology such partitions would be called exhaustive. It is natural to define the boundary of a strong (i.e., exhaustive) partition by N ( P ) := k [ i =1 ∂ Ω i ∩ Ω . When P is minimal, the set N ( P ) enjoys regularity properties which make it analogous to thenodal set of an eigenfunction of − ∆ Ω . In particular, it consists of a finite number of regularcurves. More details can be found in [HHOT09, Theorem 1.12] or [BNH17, Theorem 10.43]. Weadditionally define what it means for two distinct cluster supports Ω i and Ω j to be neighbours:the interior of the set Ω i ∪ Ω j ∩ Ω is connected. This is the analogue of Definition 2.11 inthe case of domains. We then say that the partition P is bipartite if we can colour its clustersupports, using only two colours, in such a way that two neighbours have different colours. Thisis the analogue of Definition 5.8, although in the case of domains we do not have to restrictourselves to a special class of partitions, such as proper partitions for graphs. As pointed outin [HHOT09], a nodal partition is strong and bipartite.In the case p = ∞ , which we will assume for the rest of this section, the results in [HHOT09]point to a clearer connection between minimal and nodal partitions for domains than for graphs.Indeed, [HHOT09] establishes the following result. Theorem 5.19 (Theorems 1.14 and 1.17 in [HHOT09]) . Let P be a minimal k -partition (forsome positive integer k ) realising L Dk, ∞ (Ω) . If P is bipartite, then it is nodal; more precisely, itis the nodal partition for an eigenfunction associated with λ k (Ω) , the k -th eigenvalue of − ∆ Ω . Let us recall that if ψ is an eigenfunction associated with λ k (Ω) and ν ( ψ ) its number ofnodal domains, Courant’s Nodal Theorem holds in its strong form and states that ν ( ψ ) ≤ k .Following [HHOT09], we say that ψ is Courant sharp if ν ( ψ ) = κ ( λ k (Ω)), where κ ( λ k (Ω)) = min { ℓ ∈ N ∗ ; λ ℓ (Ω) = λ k (Ω) } . The second statement in Theorem 5.19 then tells us that if a minimal k -partition P is nodal, λ k − (Ω) < λ k (Ω) and P is given by a Courant-sharp eigenfunction associated with λ k (Ω).As seen in this section, the results connecting nodal and spectral minimal partitions are lesssharp for metric graphs than for domains, except when the partitions considered are proper(see point (5) of Definition 2.12). Indeed, even when we restrict ourselves to nodal partitions,only the weak form of Courant’s Nodal Domain Theorem (Proposition 5.7) holds on graphs ingeneral. However (using the notation of this Proposition 5.7) the stronger inequality ν ( ψ ) ≤ k holds when the eigenvalue is simple or the nodal partition associated with ψ is proper.Similarly, it seems unclear how to give an appropriate definition for a bipartite partition of ametric graph that would allow us to transpose the first statement in Theorem 5.19. Nevertheless,this statement makes sense and is correct for proper minimal partitions. It is indeed formulated THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 41 as Theorem 5.9 in this section. Theorem 5.10 shows that considering only proper partitions isunnecessarily restrictive, but we do not have a comprehensive theory yet.The analogue of the second statement in Theorem 5.19 is given by Theorem 5.13 for properminimal partitions. However, there seems to be no natural way of extending the notion ofbipartiteness, and hence the scope of Theorem 5.9, to non-proper partitions, as Proposition 5.17exemplifies. Furthermore, as seen on Figure 5.1, one can also find proper minimal partitions of H , although they are not nodal. The existence of such partitions therefore does not guaranteethat L Dk, ∞ = µ k .Finally, we point out that on domains there is a construction corresponding to the realisationof proper minimal partitions as projection of nodal partitions in a double covering, describedin Theorem 5.18. This correspondence appears more clearly when we consider the magneticinterpretation of this realisation, given at the end of Subsection 5.5. As shown in [HHO13],any minimal k -partition of a domain is a nodal partition associated with the k -th eigenvalueof a magnetic Schr¨odinger operator having a finite number (possibly zero) of Aharonov–Bohmsingularities with magnetic fluxes equal to π , where the number of fluxes depends on k in arather complicated way.6. Existence of spectral maximal partitions
It turns out that for some classes of partitions the problem of maximising spectral quantitiesis also well defined: we define the energies(6.1) Ξ N ( P ) := min i =1 ,...,k µ ( G i )and(6.2) Ξ D ( P ) := min i =1 ,...,k λ ( G i )for any exhaustive rigid k -partition P ∈ R k , and thus the maximal natural and Dirichletenergies, respectively:(6.3) M Nk = M Nk ( G ) := sup P∈ R k Ξ N ( P ) , M Dk = M Dk ( G ) := sup P∈ R k Ξ D ( P ) . Here it is important to restrict to exhaustive partitions, see Remark 6.1. In the sequel we willprove similar properties of these to L Nk,p and L Dk,p , in particular the existence of maximisers.In Section 7 we will give examples comparing both the behaviour of the optimal partitionswith respect to p , and comparing these notions of spectral extremal partition with the minimalpartitions introduced in Section 4; it should be profitable to have a more systematic under-standing of the relations between them. In this section we treat the existence of partitionshaving the optimal energies for the max-min problems M Nk and M Dk . Remark 6.1.
The more general max-min problemssup P∈ A Ξ N ( P ) , sup P∈ A Ξ D ( P ) , unlike their min-max counterparts in Section 4, are only well posed for rather particular choicesof sets A of partitions. For example, if we seek the optimum among non-exhaustive k -partitions,even among rigid partitions, then both suprema are clearly infinite: any sequence of partitions P n each of whose clusters has total length at most 1 /n , say, satisfies Ξ N ( P n ) , Ξ D ( P n ) → ∞ (just use Nicaise’ inequalities).There is an analogue of Lemma 4.3, but this time for the natural problem. Lemma 6.2.
For any graph G and any k ≥ , we have sup P∈ R k Ξ N ( P ) = sup P∈ P k Ξ N ( P ) . Proof.
The proof is, to an extent, analogous to the proof of Lemma 4.3, but here we haveto prove “ ≥ ”, since the latter supremum is over a larger set. It suffices to prove that for anarbitrary exhaustive e P = { e G , . . . , e G k } ∈ P k , for each i there exists some G i ∈ ρ e Ω i (where e Ω i ⊂ G is the cluster support of e G i ) such that µ ( G i ) ≥ µ ( e G i ), since then the exhaustive rigidpartition P := {G , . . . , G k } ∈ R k satisfies Ξ N ( P ) ≥ Ξ N ( e P ). To this end, we simply take G i tobe the unique faithful cluster in ρ e Ω i ; then by construction e G i may be obtained as a cut of G i .Standard surgery results (e.g., [BKKM19, Theorem 3.4]) now imply that µ ( G i ) ≥ µ ( e G i ), asrequired. (cid:3) On the other hand, the conclusion of Remark 4.4 also holds for Ξ D ( P ): for the Dirichletproblem, there is no difference between different rigid clusters associated with the same sup-ports: in particular, maximising over all rigid partitions is equivalent to maximising over allfaithful ones.Before turning to the existence of maximising partitions, we first observe that Ξ N and Ξ D are continuous with respect to partition convergence, even in the degenerate cases. Lemma 6.3.
Suppose P n ∈ R k are rigid k -partitions of G , all similar to each other, and P n → P ∞ as n → ∞ in the sense of (3.3) . Then also Ξ N ( P n ) → Ξ N ( P ∞ ) and Ξ D ( P n ) → Ξ D ( P ∞ ) . Proof. If P ∞ is itself a k -partition, then this follows immediately from Lemma 3.6. So supposeit is not; then at least one cluster has total length converging to zero, and thus eigenvaluesdiverging to ∞ , see again Lemma 3.6. Consider the Dirichlet problem (the natural case isentirely analogous). Suppose without loss of generality that the clusters G ( ∞ )1 , . . . , G ( ∞ ) j , 1 ≤ j < n , give the minimum in Ξ D ( P ∞ ): λ ( G ( ∞ )1 ) = . . . = λ ( G ( ∞ ) j ) = Ξ D ( P ∞ );note that this energy is finite since at least one cluster of P ∞ has positive total length andthus a finite eigenvalue. But now each corresponding cluster G ( n ) i of P n converges to G ( n ) i ;in particular, λ ( G ( n ) i ) → λ ( G ( ∞ ) i ) for all i = 1 , . . . , j , while lim inf n →∞ λ ( G ( n ) i ) > Ξ D ( P ∞ )for all i = j + 1 , . . . , k . It now follows from the definition of Ξ D as a minimum that indeedΞ D ( P n ) → Ξ D ( P ∞ ). (cid:3) Theorem 6.4.
Fix a graph G and k ≥ . Then there exist exhaustive rigid partitions P N = P N ( k ) and P D = P D ( k ) of G such that Ξ N ( P N ) = M Nk and Ξ D ( P D ) = M Dk and such that there exist k -partitions P Nn , P Dn ∈ R k with P Nn → P N and P Dn → P D . For all k ≥ , P D may be taken as a k -partition itself; moreover, there exists a constant k ≥ possibly THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 43 depending on G such that P N may be taken as a k -partition for all k ≥ k . In particular, M Nk and M Dk are monotonically increasing functions of k for all k ≥ and k ≥ k , respectively. Here the idea is to apply Theorem 3.13 to the functionals Λ = − Ξ N , − Ξ D via condition (2).In the case of natural conditions, however, there is an additional difficulty with this condition;namely, it holds for all 1 ≤ ℓ ≤ k if and only if Conjecture 4.12 is true. If it is, then as we shallsee we may choose k = 1 in Theorem 6.4. Proof of Theorem 6.4.
The strong lower semi-continuity condition of Theorem 3.13 follows fromthe (strong) continuity property established in Lemma 6.3. Hence, if P Nn and P Dn are maximisingsequences of exhaustive rigid k -partitions for Ξ N and Ξ D , respectively, in both cases we obtainthe existence of exhaustive rigid limit partitions P N and P D . Although P N and P D may notbe k -partitions, Lemma 6.3 ensures that their energies are equal to M Nk and M Dk , respectively.In the Dirichlet case, we verify (2) of Theorem 3.13 to establish that either P D is already a k -partition, or it may be replaced with a rigid k -partition whose energy is no smaller. In fact, sup-pose P = {G , . . . , G ℓ } is any ℓ -partition, ℓ ≥
1, and suppose the minimum in min i =1 ,...,k λ ( G i )is achieved by G . We now modify Ω , creating an ( ℓ + 1)-st cluster support in such a way that λ ( G ) is not decreased. Take any vertex v ∈ ∂ Ω .If Ω \ { v } is disconnected, then define Ω ℓ +1 to be any one of (the closures of) these connectedcomponents and e Ω to be (the closure of) Ω \ Ω ℓ +1 . The corresponding graphs e G , G ℓ +1 maybe taken to be any graphs in the non-empty sets ρ e Ω and ρ Ω ℓ +1 , respectively.If the removal of v does not disconnect Ω , define Ω ℓ +1 to consist of exactly one edge of Ω adjacent to v and e Ω to be the rest of Ω , unless Ω consists of just one edge, in which case takeΩ ℓ +1 to consist of the half of this edge adjacent to v . The graphs are defined accordingly. Inany case, the monotonicity of the Dirichlet eigenvalues with respect to domain inclusion implies λ ( e G ) , λ ( G ℓ +1 ) ≥ λ ( G ). This establishes (2) and completes the proof of the theorem in theDirichlet case.In the natural case, it remains to establish the existence of some k ≥ k sufficiently large, if a partition realises M Nk then none of itscluster supports can wholly contain any cycle in G , and thus each cluster is a tree. We maythen apply Lemma 4.13 to subdivide these if necessary, without decreasing the energy. To thisend, we will need the following function. By way of analogy with (2.11), for any closed subsetΩ ⊂ G we define ρ Ω to be the set of all possible rigid clusters associated with Ω. We then definea function b : [0 , ∞ ) → [0 , |G| ] by(6.4) b ( λ ) := sup {| Ω | : Ω ⊂ G closed and connected and max H∈ ρ Ω µ ( H ) ≥ λ } . Now set ℓ max to be the length of the longest edge of G and c min to be the length of its shortestcycle, and choose an integer m ≥ b (cid:18) π m ℓ (cid:19) < c min . This is possible because, by Lemma 6.5 below, b ( · ) → m → ∞ . We next choose k tosatisfy m = ⌊ k /M ⌋ , where we recall M is the number of edges of G .Now fix k ≥ k . We take any (rigid) k -partition P of G in which each edge is partitionedequally into at least m clusters; there exists such a partition since m ≤ ⌊ k/M ⌋ . Then each cluster support Ω i of P is identifiable with an interval; and thus the same is true of G i . Thelongest of these has length no greater than ℓ max /m , and so M Nk ≥ min i =1 ,...,k µ ( G i ) ≥ π m ℓ . Now let P N = {G , . . . , G j k } , j k ≤ k , be an optimal partition for M Nk . Then we must have µ ( G i ) ≥ π m /ℓ for all i = 1 , . . . , j k . Hence, by choice of m and definition of b , we have |G i | < c min for all i : in particular, every cluster (and every cluster support) of P N is a tree. If P N has fewer than k of them, then we may use Lemma 4.13 to subdivide as many of the clustersof P N as necessary to create a k -partition whose energy is at least as large as Ξ N ( P N ). (cid:3) We finish by proving the properties of the function b claimed in the above proof. Lemma 6.5.
The function b : [0 , ∞ ) → [0 , |G| ] defined by (6.4) is well defined and monotoni-cally decreasing, with b ( λ ) → as λ → ∞ .Proof. The function is well defined on [0 , ∞ ) since for any λ ≥ ( n ) ⊂ G , with associated graphs G ( n ) ∈ ρ Ω ( n ) satisfying µ ( G ( n ) ) → ∞ but |G ( n ) | = | Ω ( n ) | ≥ c > n ≥
1. Since Ω ( n ) and G ( n ) are connected,the number of edges M ( G ( n ) ) of the latter is certainly not greater than 2 M (where M is thenumber of edges of the fixed graph G ), and hence, by [KKMM16, Theorem 4.2], µ ( G ( n ) ) ≤ π M ( G ( n ) ) |G ( n ) | ≤ π M |G| for all n ≥
1, a contradiction to µ ( G ( n ) ) → ∞ . (cid:3) Observe that it is easy to find a sequence of graphs G ( n ) for which µ ( G ( n ) ) → ∞ , even as |G ( n ) | remains bounded from below, if the G ( n ) are not embedded in a larger finite graph G . Remark 6.6.
Further spectral maximal partitioning problems are conceivable, in analogy tothe ones we investigated in Section 4. In particular, we may look for loose maximisers of thefunctional Ξ N ( P ) , Ξ D ( P ): we strongly expect that this problem always admits a solution, sinceeach partition is only defined by a finite number of cuts.Also, we may well introduce further functionals based on the p -means of µ ( G i ) − , ratherthen on their maximum. Again, we are confident these generalisations can be handled by thetheory developed here, but do not go into details.7. Dependence of the optimal partitions on the parameters
In the final two sections we will collect a number of miscellaneous properties of, and illus-trative examples for, the minimisation and maximisation problems from the previous sections.In this section we will consider the dependence of the two quantities L Nk,p and L Dk,p , which weconsider to be the most natural, and the partitions realising them, on p (for fixed k ), and alsoon the edge lengths of the graph G being partitioned for a fixed topology. Note that the eigenvalue numbering convention in [KKMM16] is different. Also, as observed in [BL17, § M = 2 is not correct,as any 2-flower (among certain other graphs called symmetric necklaces in [BL17]) is a maximiser in this case. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 45
Dependence on p . Let us remark that the quantities L Nk,p and L Dk,p are, for fixed G and k ≥
1, continuous and monotonically increasing in p ∈ [1 , ∞ ]. This is a general result thatfollows in exactly the same way as on domains, cf. [BNH17, Proposition 10.53]. We includethe short proof for the sake of completeness. Note that here and throughout this section, inkeeping with the convention on domains we will restrict ourselves to considering p ∈ [1 , ∞ ]. Proposition 7.1.
Fix k ≥ . For all ≤ q ≤ p ≤ ∞ , we have L Nk,q ( G ) ≤ L Nk,p ( G ) ≤ k q − p L Nk,q ( G ) and L Dk,q ( G ) ≤ L Dk,p ( G ) ≤ k q − p L Dk,q ( G ) (where /p = 0 if p = ∞ ). Consequently, the mappings p
7→ L
Nk,p ( G ) and p
7→ L
Dk,p ( G ) arecontinuous and monotonically increasing in p ∈ [1 , ∞ ] .Proof. We give the proof for L Nk,p ; the proof for L Dk,p is identical. In fact, it suffices to show that(7.1) Λ Nq ( P ) ≤ Λ Np ( P ) ≤ k q − p Λ Nq ( P )for any rigid k -partition P , since then the same is true for the corresponding infima over allsuch partitions. But (7.1) is a direct consequence of the H¨older inequality, using the definition(4.1) of Λ Np ( P ). (cid:3) We continue discussing the dependence of optimal partitions and energies on p . To begin with,let us present a concrete example illustrating how the optimal partition, say in the simplest casefor L D ,p , can depend nontrivially on p . On domains, relatively little seems to be known, and mostof the work to date seems to have been of (largely) numerical nature; see in particular [BBN18].Our example, in addition to establishing that L D ,p and the corresponding optimal partitionscan, in fact, depend on p , should also demonstrate how in the case of metric graphs it seemspossible to prove more properties (such as monotonicity of the deformation in p ) analytically.On the other hand, since L Dk, ∞ need not be realised by an equipartition, it follows that knowncriteria for establishing the inequality L Dk, < L Dk, ∞ (see [BNH17, Proposition 10.54] or [HHO10])have no direct equivalent; see Proposition 7.8 and the discussion around it. Moreover, strictinequality here is possible even if the optimal partition is independent of p ; see Example 7.6. Example 7.2.
We return to the equilateral star graph G on three edges of length 1 each,considered in Example 4.10, and ask for the partitions achieving L D ,p for p ∈ [1 , ∞ ) (we recallthat when p = ∞ , up to isometry there is one optimal partition, whose cut set consists of (only)the central vertex of degree 3). Any 2-partition P = {G , G } of G may, up to symmetries, beidentified uniquely by the location of its cut set v along a given, fixed edge (see Figure 7.1). v a G G Figure 7.1.
The equilateral 3-star G . The white circle denotes the cut set { v } of the two-partition P = {G , G } ; the corresponding edge is divided into piecesof length a in G and 1 − a in G . Since the edge has length 1, the partition may uniquely be described by a single parameter a ∈ [0 , a = 0 corresponding to a cut in the central vertex and thus the partitionrealising L D , ∞ , and a = 1 formally giving G = G and G = { v } . The following propositiondescribes how the optimal cut point depends on p . We will give its proof below. Proposition 7.3.
For the equilateral 3-star G , in the notation and setup of Example 7.2, foreach p ∈ [1 , ∞ ] , there is a unique value of a ∈ [0 , whose corresponding partition achieves L D ,p ( G ) , which we denote by a p . Then a p is a smooth function of p , with a p > and ddp a p < for all p ∈ [1 , ∞ ) , and lim p →∞ a p = 0 . In particular, the optimal partition is never the equipartition except for p = ∞ (which werecall corresponds to a ∞ = 0), and the cut point v , as a function of p , moves smoothly andmonotonically from its location at p = 1 towards the central vertex as p → ∞ . This mirrors verymuch the numerically observed behaviour of the (conjectured) optimal partitions on domainsin [BBN18]. Remark 7.4.
As an immediate consequence of Proposition 7.3, we obtain the inequalities µ ( G ) = L D , ∞ ( G ) > L D ,p ( G )for all p ∈ [1 , ∞ ), for the example where G is an equilateral 3-star. In particular, there is nogeneralisation of Proposition 5.5 to p = ∞ .The above example suggests that the optimal partition for L Dk,p ( G ) should depend on p whenever there is an optimal partition for p = ∞ which is not internally connected . It wouldtake us too far afield to consider this question here, so we formulate it as an open problem. Conjecture 7.5.
Let G be given and let k ≥ k -partition P achieving L Dk, ∞ ( G ) which is notinternally connected. Then P does not achieve L Dk,p ( G ) for any p < ∞ , that is, Λ Dp ( P ) > L Dk,p ( G ) for all p < ∞ .(2) Whenever there exists a rigid k -partition P achieving L Dk, ∞ ( G ) but which, for some p < ∞ , does not achieve L Dk,p ( G ), then we have that L Dk,p ( G ) is a strictly monotonicfunction of p . Proof of Proposition 7.3.
First we compute the energy as a function of a ∈ [0 , Dp ( a ) := Λ Dp ( {G ( a ) , G ( a ) } )We clearly have λ ( G ( a )) = π / − a ) . Noting that the eigenfunction is identical on the twoidentical edges, we can obtain that λ ( G ) =: ω ( a ) , with ω ( a ) the smallest positive solutionof the secular equation(7.2) 2 tan( aω ) = cot( ω ) . We note that ω ( a ) < π/ a ∈ (0 ,
1) and, by implicit differentiation,(7.3) dωda = − ωa + cos ( aω )2 sin ( ω ) . Let us now show that, for all a ∈ (0 , d ωda ( a ) > . THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 47
Differentiating equation (7.3), we find d ωda ( a ) = ω (cid:16) a + cos ( aω )2 sin ( ω ) (cid:17) + ω (cid:16) a + cos ( aω )2 sin ( ω ) (cid:17) (cid:18) aω )sin( ω ) dda (cid:18) cos( aω )sin( ω ) (cid:19)(cid:19) . Using equation (7.3) again, we obtain dda (cid:18) cos( aω )sin( ω ) (cid:19) = 1sin ( ω ) (cid:18) − (cid:18) ω + a dωda (cid:19) sin( aω ) sin( ω ) − dωda cos( aω ) cos( ω ) (cid:19) = ω sin ( ω ) (cid:16) a + cos ( aω )2 sin ( ω ) (cid:17) (cid:18) a sin( aω ) sin( ω ) + cos( aω ) cos( ω ) − (cid:18) a + cos ( aω )2 sin ( ω ) (cid:19) sin( aω ) sin( ω ) (cid:19) = ω cos( aω )2 sin ( ω ) (cid:16) a + cos ( aω )2 sin ( ω ) (cid:17) (2 cos( ω ) sin( ω ) − cos( aω ) sin( aω )) . We have, using the secular equation (7.2),2 cos( ω ) sin( ω ) − cos( aω ) sin( aω ) = 2 cot( ω ) sin ( ω ) − tan( aω ) cos ( aω )= (4 sin ( ω ) − cos ( aω )) tan( aω ) . Using again the secular equation, we find successively4 tan ( aω ) = cot ( ω );4 (cid:18) ( aω ) − (cid:19) = 1sin ( ω ) − ( aω ) − ( ω ) = 3 . It follows that 4 sin ( ω ) − cos ( aω ), and therefore also dda (cid:16) cos( aω )sin( ω ) (cid:17) and d ωda ( a ), are positive.For convenience, we define the function F ( a, p ) := 2Λ Dp ( a ) p = (cid:18) π − a ) (cid:19) p + ω ( a ) p . We have immediately(7.5) ∂F∂a ( a, p ) = 2 p (cid:18)(cid:16) π (cid:17) p − a ) p +1 + ω p − dωda (cid:19) . We have ω (0) = π and so, using equation (7.3), ∂F∂a (0 , p ) = 2 p (cid:18)(cid:16) π (cid:17) p − (cid:16) π (cid:17) p (cid:19) = − p (cid:16) π (cid:17) p < , so that a = 0 is not even a local minimum of a F ( a, p ); while, as a → (cid:16) π (cid:17) p p (1 − a ) p +1 −→ + ∞ and the term ω p − dωda = − pω p a + cos ( aω )2 sin ( ω ) is bounded. We conclude ∂∂a F ( a, p ) → + ∞ as a →
1. On the other hand, ∂ F∂a ( a, p ) = 2 p (cid:16) π (cid:17) p p + 1(1 − a ) p +2 + (2 p − ω p − (cid:18) dωda (cid:19) + ω p − d ωda ! , which is clearly positive as a consequence of inequality (7.4). The function a ∂∂a F ( a, p ) istherefore increasing, and has a unique zero in [0 , a F ( a, p ). We have proved the first part of Proposition 7.3.Since ∂ ∂a F ( a p , p ) >
0, it follows from the Implicit Function Theorem that p a p is contin-uously differentiable (indeed, even real analytic) and that da p dp ( p ) = − ∂ ∂p∂a F ( a p , p ) ∂ ∂a F ( a p , p ) . Differentiating equation (7.5) with respect to p , we find ∂F∂p∂a ( a, p ) = 1 p ∂F∂a ( a, p ) + 2 p (cid:18) (cid:18) π − a ) (cid:19) (cid:16) π (cid:17) p − a ) p +1 + 2 log( ω ) ω p − dωda (cid:19) . Using ∂∂a F ( a p , p ) = 0 and equation (7.5), we obtain in particular ∂ F∂p∂a ( a p , p ) = 8 p (cid:16) π (cid:17) p − a p ) p +1 log (cid:18) π − a p ) ω ( a p ) (cid:19) . Since a p ∈ (0 ,
1) and ω ( a p ) ∈ (0 , π/ p a p decreasing.As a positive and decreasing function defined on [1 , ∞ ), p a p has a non-negative limit at ∞ , which we denote by a ∗ . Let us assume by contradiction that a ∗ >
0. Using equations (7.3)and (7.5), the condition ∂∂a F ( a p , p ) = 0 can be written (cid:16) π (cid:17) p − a p ) p +1 = ω ( a p ) p a p + cos ( a p ω ( a p ))2 sin ( ω ( a p )) . It follows that π − a p ) (1 − a p ) − p = ω ( a p ) (cid:18) a p + cos ( a p ω ( a p ))2 sin ( ω ( a p )) (cid:19) − p . Passing to the limit p → ∞ , we obtain π − a ∗ ) = ω ( a ∗ ) , in contradiction to ω ( a ∗ ) < π/
2. We conclude that a ∗ = 0 = a ∞ . (cid:3) We now return to the meaning of the inequality L Dk, ( G ) < L Dk, ∞ ( G ) for a metric graph G . Example 7.6.
We give a simple example where the optimal partition for L Dk,p ( G ) is independentof p ∈ [1 , ∞ ] but the optimal energy L Dk,p ( G ) itself is not; this is a direct consequence of theexistence of certain minimal partitions which are not equipartitions. Indeed, take G to be a notquite equilateral star on three edges, say of length | e | = 1 + ε , | e | = | e | = 1. We denote by v THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 49 the central vertex of G and by v , v and v the pendant vertices of e , e and e respectively.Let us denote by P the 3-partition of G whose cut set is v . ThenΛ Dp ( P ) = (cid:18) (cid:18) π p (2 + 2 ε ) p + 2 · π p p (cid:19)(cid:19) /p . This energy clearly depends on p . The following proposition establishes that our example hasthe desired properties. Proposition 7.7.
There exists ε > such that, for all ε ∈ [0 , ε ] and all p ∈ [1 , ∞ ] , P is theunique -partition realising L D ,p ( G ) .Proof. Let us first prove the proposition for p ∈ [1 , ∞ ), by a straightforward discussion of thepossible topological cases. Let us consider an exhaustive 3-partition P = {G , G , G } different from P . Its cut set { w , w } consists of two points, each distinct from the centralvertex. This leaves us with four essentially distinct cases (all the other cases reduce to thesefour by relabelling of the edges and cut points):(i) the two points belong to e ;(ii) the two points belong to e ;(iii) w belongs to e and w to e ;(iv) w belongs to e and w to e .In all cases, we denote by a i the distance of w i from v , i = 1 ,
2. To simplify notation, we define F ( p ) := 3 · p π p Λ Dp ( P ) p = 1(1 + ε ) p + 2 . In Case (i), up to relabelling, we can assume that a < a . We can also assume that G is the 3-star with edges e , e and [ v, w ], with a boundary condition at each pendant vertexrespectively Neumann, Neumann and Dirichlet. We recall that here and in the rest of the proofwe have a natural boundary condition at the central vertex v . We can finally assume that G is the segment [ w , w ] with Dirichlet boundary conditions and G the segment [ w , v ] witha Dirichlet boundary condition at w and a Neumann boundary condition at v . We have,recalling that p ≥ · p π p Λ Dp ( P ) p = (cid:18) π λ ( G ) (cid:19) p + (cid:18) π λ ( G ) (cid:19) p + (cid:18) π λ ( G ) (cid:19) p = (cid:18) π λ ( G ) (cid:19) p + 2 p ( a − a ) p + 1(1 − a ) p > p + 1 ≥ > > F ( p ) . In Case (ii), we assume, without loss of generality, that a < a , G is the three star with edges e , e and [ v, w ], with a boundary condition at each pendant vertex respectively Neumann,Neumann and Dirichlet, G is the segment [ w , w ] with Dirichlet boundary conditions and G the segment [ w , v ] with a Dirichlet boundary condition at w and a Neumann boundarycondition at v . We have3 · p π p Λ Dp ( P ) p = (cid:18) π λ ( G ) (cid:19) p + 2 p ( a − a ) p + 1(1 + ε − a ) p > p ( a − a ) p > p (1 + ε ) p . If we assume ε ≤ / √ − <
1, it follows, since p ≥ · p π p Λ Dp ( P ) p > p (1 + ε ) p ≥ (1 + ε ) ≥ > F ( p ) . In Case (iii), we assume, without loss of generality, that G is the 3-star with edges e , [ v, w ]and [ v, w ], with boundary conditions respectively Neumann, Dirichlet and Dirichlet at thependant vertices. We also assume that G and G are respectively the segments [ w , v ] and[ w , v ] with Dirichlet-Neumann boundary conditions. To simplify notation, we introduce F ( a , a ) := 3 · p π p Λ Dp ( P ) p = (cid:18) π ω ( a , a ) (cid:19) p + 1(1 − a ) p + 1(1 − a ) p , where ω ( a , a ) is the smallest positive solution of the equation(7.6) cotan( a ω ) + cotan( a ω ) = tan( ω ) , so that λ ( G ) = ω ( a , a ) . To simplify notation, we write ω := ω ( a , a ). We now showthat ∂∂a F ( a , a ) is positive for all ( a , a ) ∈ (0 , F ( a , a ) > lim a → F ( a , a ). Noticing thatlim a → F ( a , a ) = 1(1 + ε ) p + 1(1 − a ) p + 1 > ε ) p + 2 = F ( p ) , we conclude that the claim implies Λ Dp ( P ) > Λ Dp ( P ). Let us now prove the claim. By inspectionof Equation (7.6), it is clear that ω ∈ (0 , π/ π/
2, so that a ω and a ω also belong to (0 , π/ a and simplifying,we find(7.7) ∂ω∂a = − ωa + (cid:0) sin a ω cos ω (cid:1) + (cid:16) sin a ω sin a ω (cid:17) . To go further, let us note that since ω , a ω and a ω belong to (0 , π/ < cotan( a ω ) < tan( ω ). Taking the square, we find1(sin a ω ) = cotan( a ω ) < tan( ω ) = 1(cos ω ) , that is to say (sin a ω ) (cos ω ) > . It follows that(7.8) (cid:12)(cid:12)(cid:12)(cid:12) ∂ω∂a (cid:12)(cid:12)(cid:12)(cid:12) < ω < π . Using Inequality (7.8), we now find ∂F∂a ( a , a ) = 2 p (cid:18) π (cid:19) p ∂ω∂a ω p − + 1(1 − a ) p +1 ! ≥ p − a ) p +1 − (cid:18) π (cid:19) p (cid:12)(cid:12)(cid:12)(cid:12) ∂ω∂a (cid:12)(cid:12)(cid:12)(cid:12) ω p − ! > , THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 51 proving the claim. Let us note that the derivative of F ( a , a ) with respect to a is also positive,by symmetry.Let us finally study Case (iv). We assume, without loss of generality, that G is the 3-starwith edges e , [ v, w ] and [ v, w ], with boundary conditions respectively Neumann, Dirichlet andDirichlet at the pendant vertices and that G and G are respectively the segments [ w , v ] and[ w , v ] with Dirichlet-Neumann boundary conditions. Assuming that a ≤
1, we can repeatthe computation of Case (iii) and show that ∂∂a F ( a , a ) is positive, and therefore3 · p π p Λ Dp ( P ) p = F ( a , a ) > lim a → F ( a , a ) > F ( p ) . If a > ε ≤ / √
2, we have3 · p π p Λ Dp ( P ) p > (cid:18) π λ ( G ) (cid:19) p + (cid:18) π λ ( G ) (cid:19) p = 1(1 + ε − a ) p + 1(1 − a ) p > ε p + 1 ≥ > F ( p ) . Altogether, assuming that ε ≤ ε := min (cid:0) / √ − , / √ (cid:1) = 2 / √ −
1, Λ Dp ( P ) > Λ Dp ( P )in all cases. By a limiting argument, we could show immediately that P is a minimal partitionrealising L D , ∞ . This would however not prove uniqueness. It is better to give a direct proof.We first note that Λ D ∞ ( P ) = π /
4. On the other hand, let us again consider P , an exhaustive3-partition distinct from P . In Cases (i), (iii) and (iv), P has a domain strictly containedin an edge of length 1. In case (iii), assuming ε ≤
1, the edge e contains a segment withDirichlet-Neumann conditions and one with Dirichlet-Dirichlet conditions, one of which haslength less than 1. It follows that, in all cases, Λ D ∞ ( P ) > π / (cid:3) The value ε = 2 / √ − ε cannot bechosen too large. More explicitly, if ε > ε := 2 √ −
1, and if we choose for P a partition of type(iii) (see the proof of Proposition 7.7) such that a − a > ε ) / ε − a > (1 + ε ) / Dp ( P ) < Λ Dp ( P ) for all p ∈ [1 , ∞ ].A “balancing formula” which can be found in [BNH17, Proposition 10.54] gives a sufficientcondition under which the inequality L Dk, < L Dk, ∞ holds for domains Ω ⊂ R . For a k -partition P = { Ω , . . . , Ω k } of Ω, minimal for L Dk, ∞ , if Ω i and Ω j are neighbours and Ω ij is the interiorof the closure of Ω i ∪ Ω j , then(7.9) λ (Ω i ) = λ (Ω j ) = λ (Ω ij ) = L Dk, ∞ (Ω);if ψ i and ψ j are the respective eigenfunctions on Ω i and Ω j , extended by zero to the rest ofΩ, scaled in such a way that ψ i + ψ j is an eigenfunction for Ω ij ; then the formula states that L Dk, (Ω) < L Dk, ∞ (Ω) if, under this normalisation,(7.10) k ψ i k L (Ω i ) = k ψ j k L (Ω j ) . This may fail on graphs, as Example 7.6 shows. However, we still have the following positiveresult, based on a slightly different normalisation.Here we assume for fixed k that P = {G , . . . , G k } is a k -partition that achieves L Dk, ∞ ( G ), andwe suppose G i and G j to be any two neighbours (in the sense of Definition 2.11), with respectiveeigenfunctions ψ i and ψ j . We treat ψ i and ψ j as elements of H ( G ), extending them by zerooutside Ω i and Ω j , respectively. Proposition 7.8.
Under the conditions described above, if v ∈ ∂ Ω i ∩ ∂ Ω j is a vertex of degreetwo in G and ψ i and ψ j are normalised in such a way that ψ i + ψ j is continuously differentiableat v , then condition (7.10) implies that L Dk, ( G ) < L Dk, ∞ ( G ) . The idea of the proof, based on a formula of
Hadamard type for domain perturbation (see,for example, [BKKM19, Remark 3.14] for the metric graph version), is essentially identical ongraphs, and we omit it.7.2.
Dependence on the edge lengths.
Here we give an example which shows that L N , ∞ ,while still continuous, need not be a smooth function of the edge lengths of the underlyinggraph. This is caused by the existence of an isolated value of the length of a given edge forwhich there is non-uniqueness of the minimiser, and a transition from one type of minimiserto another at this point: one family of partitions “leapfrogs” the other. It would be interestingto know whether a similar phenomenon is possible if, instead of varying p , we vary the edgelengths. Example 7.9.
We consider the lasso graph of Figure 2.1; we will denote by G a the particularlasso graph for which | e | = | e | = 1 but | e | = a ∈ [2 , ∞ ). We distinguish four cases:(1) a = 2. Here the optimal partition for L N , ∞ ( G ) = π / a ∈ (2 , L N , ∞ ( G a ) is still the partitionfrom Figure 2.4, meaning in particular that L N , ∞ ( G a ) = π / a ∈ [2 , a ∈ (2 , b ∈ (0 , a = 3. Here we have two minimisers, depicted in Figure 7.2, and L N , ∞ ( G ) = π / e , as otherwise the right-hand cluster would have an eigenvalue largerthan π /
4. But the strict dependence of the eigenvalue of an interval on the length ofthe intercal, plus the strict monotonicity statement of Lemma 7.10, guarantees that anyother partition of G which cuts through e will have a higher energy than π / π / π / π / π / e e e Figure 7.2.
Two different partitions realising L N , ∞ ( G ), together with the re-spective eigenvalues of the partition clusters. On the left the original edges of G are labelled; while on the right the edge lengths for the corresponding partitionclusters are displayed.(4) a ∈ (3 , ∞ ). Here there must be a unique minimal partition, which is a smooth evolutionof the one depicted on the right of Figure 7.2 as a > a . This follows, firstly, from the necessity of the partition being an equipartition,and the strictly negative derivative of the eigenvalues with respect to the edge lengths(use [BKKM19, Remark 3.14]). THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 53
The optimal partition energy L N , ∞ ( G a ) is thus a continuous and (weakly!) monotonically de-creasing, but not C , function of a ∈ [2 , ∞ ). Lemma 7.10.
The lasso graph G b of Figure 2.1, with side lengths | e | = | e | = 1 and | e | = b > , has the following properties:(1) µ ( G b ) is simple and a smooth, strictly monotonically increasing function of b > ;(2) µ ( G ) = π / .In particular, µ ( G b ) > π / whenever b < .Proof. (1) is an immediate consequence of [BKKM19, Lemma 5.5 and Corollary 3.12(1)], to-gether with the well-known analyticity of the eigenvalues in dependence on the edge lengths(see, e.g., [BK12] or [BK13, § G with the equilateral 3-star G whose edges are each of length one. Now G can be realised from G by cutting through thevertex z ; moreover, since there exists an eigenfunction for µ ( G ) which takes on the samevalue at the two degree one vertices of G which can be glued together to obtain G , we have µ ( G ) = µ ( G ) = π / (cid:3) Comparison of different partition problems
In this section we will compare different types of partitions problems and their correspondingoptimal partitions on a fixed graph, mostly via illustrative examples. We will mostly restrictourselves to the case of rigid partitions and in particular to the four quantities L Nk,p , L Dk,p , M Nk and M Dk . We will, however, also look briefly at properties of non-exhaustive partitions and inparticular consider their relationship to non-rigid partitions.8.1. Comparison of L Nk,p , L Dk,p , M Nk and M Dk . Here we give a few prototypical examplesillustrating how these four problems give rise to different optimal partitions, and compare theactual optimal energies. In terms of the form of the optimal partitions, the examples provideat least preliminary evidence to suggest that (very roughly speaking) L Nk,p tends to seek out thelongest possible paths within the graph; L Dk,p (and possibly also M Nk ) tends to divide the graphinto k roughly equal pieces, preserving highly connected parts of the graph; while the optimalpartition for M Dk tends to cut through the highly connected parts. It would be worthwhile toinvestigate this systematically and try to provide a rigorous basis for these heuristic claims.We start with a simple criterion for identifying optimal partitions for the Neumann problemin some cases. Lemma 8.1.
Suppose G has total length L > and A is any set of k -partitions of G . Supposethere exists a partition P ∗ ∈ A such that Λ N ∞ ( P ∗ ) = π k L . Then (8.1) Λ N ∞ ( P ∗ ) = inf (cid:8) Λ N ∞ ( P ) : P ∈ A (cid:9) , P ∗ is exhaustive, and the clusters G , . . . , G k of P ∗ are all path graphs (that is, intervals) oflength L/k . In particular, P ∗ is also a minimising partition for e L Nk,p , for all p ∈ (0 , ∞ ] . We will generally take A = R k to be the set of exhaustive rigid k -partitions, in which casethe infimum in (8.1) is, by definition, L Nk, ∞ , but the result holds for any set A of k -partitions.There is also a corresponding statement for Dirichlet minimal k -partitions, namely that the same conclusion holds if Λ D ∞ ( P ∗ ) = π k / (4 L ), but in this case each cluster must be a pathgraph of length L/k with a Dirichlet vertex at one endpoint, which in particular requires G tobe an equilateral k -star. Proof of Lemma 8.1.
Let
P ∈ A be a k -partition with clusters G , . . . , G k . Then the sharp formof Nicaise’ inequality (Theorem 2.5) implies that µ ( G i ) ≥ π / |G i | for all i = 1 , . . . , k , withequality if and only if G i is a path graph. The claim now follows immediately from the definitionof Λ N ∞ . (cid:3) Example 8.2.
Consider the graph G depicted in Figure 8.1, consisting of two longer andfour shorter edges, the former of length 1 and the latter of length a , which we imagine to beconsiderably smaller than 1. One may visualise G as a loop with two short equal “reinforcing”edges placed at antipodal points. On G , we will consider the respective partitions achieving thefour quantities L D , ∞ , L N , ∞ , M D and M N ; by symmetry considerations, the two clusters mustalways have equal energies and hence be congruent to each other. Since the purpose of thisexample is essentially heuristic, we will not give detailed proofs of our claims. This would bepossible, albeit tedious, via case-bashing arguments based for example on the advanced surgerytechniques of [BKKM19] together with extensive use of the symmetry of the problems. Figure 8.1.
The graph G . We assume the longer edges both have length 1, andthe four shorter edges all have the same length a < L D , ∞ , we are seeking the nodal partition corresponding to µ ( G ) (see Proposition 5.6). Anargument similar to the ones in [BKKM19, § µ ( G ) mustbe invariant with respect to permutation of the longer two edges, and of each of the shorterones within each set of two adjacent short edges. This leaves only the possibilities depictedin Figure 8.2 (left) and Figure 8.3 (left). Either a direct calculation involving the secularequations or an argument analogous to [BKKM19, Proposition 5.10] shows that the former haslower energy and hence corresponds to µ and L D , ∞ . For L N , ∞ , we see that it is possible topartition G into two equal path graphs as shown in Figure 8.2 (right); hence, by Lemma 8.1,this is the optimal partition. THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 55
Figure 8.2.
Rigid 2-partitions of G realising L D , ∞ ( G ) (left) and L N , ∞ ( G ) = e L N , ∞ ( G ) (middle). There exists a further loose partition realising e L N , ∞ ( G ) (right).Black dots denote Neumann-Kirchhoff vertices, white dots denote Dirichlet ver-tices.If we turn to maximal partitions, in the Dirichlet case the same argument as in [BKKM19,Proposition 5.10] implies that among all Dirichlet candidates, the eigenvalue λ is largest whenthe shorter edges are as close to the Dirichlet (cut) set and as equal as possible, yielding Fig-ure 8.3 (left). To maximise the Neumann eigenvalues, the doubled edges should be located asclose to, and as symmetrically about, the zero set of the eigenfunctions as possible, correspond-ing to Figure 8.3 (right). We omit the details, which closely follow the principles laid out in[BKKM19, § Figure 8.3.
Proper partitions of G realising M D ( G ) (left) and M N ( G ) (right). Example 8.3.
Our second example will be the equilateral -pumpkin H depicted in Figure 8.4:it is the graph with two vertices and six parallel edges running between them, all of the samelength (say, length 1 each). Figure 8.4.
An equilateral “6-pumpkin” (left) and a rigid partition realising L N , ∞ on it (right). Again, no loose partition achieves an energy lower than L N , ∞ .In Figure 8.4 we also see how H can be partitioned into two path graphs of length 3 each;by Lemma 8.1, this is an optimal 2-partition, corresponding to L N , ∞ ( H ). This is thus another illustration of the assertion that “ L Nk, ∞ tends to seek out paths embedded in the graph”. Interms of Dirichlet minimal partitions, to find partitions achieving L D , ∞ ( H ), by Proposition 5.6we merely have to consider the nodal patterns of eigenfunctions corresponding to µ ( H ); theseare depicted in Figure 8.5. We observe explicitly that this is an (easy) example of an intrinsicnon-uniqueness: there are two completely different partitions of H which both yield L D , ∞ ( H ). Figure 8.5.
Two different partitions realising L D , ∞ on the equilateral 6-pumpkin. The partition on the left is internally connected, the one on the rightis not as the Dirichlet vertices disconnect the clusters. Example 8.4.
Let us finally present an example of a graph with different rigid and looseminimal partitions: an equilateral dumbbell graph G . The unique minimiser for L N , ∞ ( G ) andthe different minimiser for e L N , ∞ ( G ) are shown in Figure 8.6. The former is also a minimiser for L D ,p ( G ), p ∈ (0 , ∞ ]. Figure 8.6.
A dumbbell graph, its rigid Neumann/Dirichlet minimal partitionand a loose Neumann minimal partition8.2.
Exhaustive versus non-exhaustive partitions.
We now give a few remarks on whatcan be expected if we allow, or disallow, non-exhaustive partitions.
Example 8.5.
We give a basic example to show that if instead of L Nk,p we consider the corre-sponding minimisation problem among all non-exhaustive rigid partitions, then there may beno exhaustive minimising partition. If we take k = 1 and any p ∈ (0 , ∞ ], then the claim is thatthere exist a graph G and a proper subset (i.e. a non-exhaustive one-partition) Ω ⊂ G , as wellas a graph G associated with Ω , such that µ ( G ) < µ ( G ). Take G to be a loop of length 1and Ω to be a connected subset of it of length 1 − ε , so that G is a path graph of length 1 − ε .Then µ ( G ) = π (1 − ε ) < π = µ ( G )as long as ε ∈ (0 , / P := {G} is not the optimal 1-partition of G if non-exhaustive partitions are allowed. Example 8.6.
Here is another example akin to Example 8.5, but where we can more clearly seethe similar role to non-exhaustive partitions played by non-rigid (but exhaustive) ones. We take G to be an equilateral pumpkin graph on three edges of length one each (cf. Example 8.3) and,analogously to the previous example, seek a 1-partition minimising L N ,p , p ∈ (0 , ∞ ]. Allowing THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 57 exhaustive loose partitions, we can cut through the vertices of G to produce a path graph I ,much as was done in Figure 8.4 (right); see Figure 8.7 (left). Lemma 8.1 (with A = P k ) showsthat P ∗ = {I} , and not P = {G} , is the optimal partition for e L N ,p ( G ), p ∈ (0 , ∞ ]. Indeed, theformer has energy Λ Np ( P ∗ ) = µ ( I ) = π /
9, while the latter has energy Λ Np ( P ∗ ) = µ ( G ) = π .11 1 11 − ε εε − ε Figure 8.7.
A minimal 1-partition of the 3-pumpkin of Example 8.6 amongloose but exhaustive partitions (left) and a non-exhaustive but rigid 1-partitionapproximating it (right; the dotted lines represent the parts of the graph excludedfrom the cluster support). The numbers give the lengths of the respective edges.But now consider rigid non-exhaustive 1-partitions of G . By removing an arbitrarily smallpiece of edge near each of the vertices, we can produce a partition consisting of a single pathgraph of length arbitrarily close to 3; see Figure 8.7 (right). As the piece removed becomessmaller, we have convergence of the length to 3, and thus convergence of the eigenvalue (equally,the energy of the partition) to π / Np ( P ∗ ).The preceding example suggests that in the natural case, infimising over non-exhaustivepartitions is equivalent to minimising over loose exhaustive partitions. Indeed, whenever, onforming an optimal partition, one wishes to detach a single edge from a vertex (as was donein Example 8.6), one can do this either by allowing loose partitions or infimising over non-exhaustive ones. At work here are two “surgery” principles: cutting through vertices decreases µ , as does lengthening pendant edges (see, e.g., [KKMM16, Lemma 2.3]). However, in generalthe two are not equivalent, as Example 8.8 below shows. Since allowing loose partitions ingeneral seems to give more freedom to cut through vertices than allowing non-exhaustive rigidones, we still expect the following general principle to hold. Since it would take us too far afieldto prove the claim in the current context, we record it as a conjecture for future work. Conjecture 8.7.
Fix a graph G , a number k ≥ p ∈ (0 , ∞ ]. Then the quantity(8.2) inf (cid:8) Λ Np ( P ) : P is a non-exhaustive but rigid k -partition of G (cid:9) is no smaller than(8.3) e L Nk,p = inf (cid:8) Λ Np ( P ) : P is an exhaustive loose k -partition of G (cid:9) . Example 8.8.
We sketch an example of a graph G , with k = 2 and p = ∞ , where the quantityin (8.3) is strictly smaller than the one in (8.2), which in turn is strictly smaller than L N , ∞ ( G ).We take G to be the “dumbbell”-type graph depicted in Figure 8.8, consisting of a bridge(“handle”) e of length 1, with a chain of two loops of total length ε > e ; thus G has total length 1 + 2 ε . e ε ε Figure 8.8.
A dumbbell with “double weights”. The handle e has length 1and the total length of each of the figure-8 “weights” is ε .Firstly, a short symmetry argument shows that L N , ∞ ( G ) is given by the partition bisecting G at the midpoint of e .Secondly, if we allow loose partitions, then we can partition G into two equal path graphs oflength 1 / ε each, as depicted in Figure 8.9 (left). Since this gives the smallest possible energyfor a 2-partition, by Lemma 8.1 it yields the minimum, and the minimum is only achieved bytwo equal path graphs. (In particular, e L N , ∞ ( G ) < L N , ∞ ( G ).) Figure 8.9.
The optimal partition among all loose exhaustive ones (left) and acandidate for the infimum among all rigid but non-exhaustive ones (right).Finally, we consider rigid but non-exhaustive partitions. In this case, a topological argumentshows that it is impossible to obtain two equal path graphs, since there is no way to cut througha degree-four vertex to obtain two degree-two vertices within a cluster support. Thus thereis strict inequality between (8.2) and (8.3) in this case. However, if we take the partitionin Figure 8.9 (right), which is achievable as the limit of a sequence of non-exhaustive rigidpartitions, then this partition is obtainable from the optimal one for L N , ∞ ( G ) by cutting throughcertain vertices of the latter. One may then show using the strictness statement in [BKKM19,Theorem 3.4] (in the form of Remark 3.5) that this partition has strictly lower energy than L N , ∞ ( G ); thus the infimum (8.2) is also lower. This completes the proof of the claimed chain ofstrict inequalities.We shall finish with a different observation on non-exhaustive partitions. We saw in Exam-ple 5.12 that a partition P ∗ achieving L Dk, ∞ need not be an equipartition. However, if we allownon-exhaustive partitions, then we can always “artificially generate” a minimal equipartition byshrinking every cluster in P ∗ whose first eigenvalue is too large. Similarly, by then discardingsuperfluous connected components in each cluster, we can guarantee that the resulting minimalequipartition consists only of internally connected clusters. Proposition 8.9.
Suppose P = {G , . . . , G k } is an (exhaustive) rigid k -partition of G suchthat Λ D ∞ ( P ) = L Dk, ∞ ( G ) for some k ≥ . Then there exists another, possibly non-exhaustive butequilateral rigid k -partition P ′ = {G ′ , . . . , G ′ k } such that also Λ D ∞ ( P ′ ) = L Dk, ∞ ( G ) . The partition P ′ may additionally be chosen in such a way that the clusters G ′ , . . . , G ′ k are all internallyconnected.Proof. It suffices to prove that if H is any graph with a non-empty set of Dirichlet vertices V D ( H )and λ ≥ λ ( H ; V D ( H )), then there exists a subgraph e H ⊂ H with boundary (equivalently,Dirichlet vertex set)(8.4) ∂ e H := (cid:16) e H ∩ H \ e H (cid:17) ∪ (cid:16) e H ∩ V D ( H ) (cid:17) THEORY OF SPECTRAL PARTITIONS OF METRIC GRAPHS 59 such that λ ( e H ; ∂ e H ) = λ . Indeed, in this case, whenever λ ( G i ) < Λ D ∞ ( P ), we simply findsome G ′ i ⊂ G i such that λ ( G ′ i ) = Λ D ∞ ( P ).To this end, simply choose any fixed z ∈ V D ( H ) and for t ∈ [0 , max x ∈H dist( x, z )] set H t := H \ { x ∈ H : dist( x, z ) < t } (where dist denotes the Euclidean distance within H , defined such that paths may not passthrough vertices in V D ( H ), and ∂ H t is defined as in (8.4)). We claim that t λ ( H t ; ∂ e H t )is continuous for t ∈ [0 , max x ∈H dist( x, z )] (although H t may not necessarily be connected).Assuming this claim, since also |H t | → t → max x ∈H dist( x, z )], we also have λ ( H t ; ∂ e H t ) →∞ ) by [Nic87, Th´eor`eme 3.1]. The statement of the proposition thus follows.To prove the claim, we simply observe that this is a special case of Lemma 3.6, where thegraphs need not be connected. Indeed, fix t ∈ [0 , max x ∈H dist( x, z )] and a sequence t n → t ;then the conclusion follows with H t = G ∞ and H t n = G n .Finally, fix an arbitrary cluster G ′ i and suppose that it is not internally connected. Withoutloss of generality, we regard all points in ∂ G ′ i , which are equipped with a Dirichlet condition,as having degree one: then in particular G ′ i is not connected. Let ψ i be any eigenfunctioncorresponding to λ ( G ′ i ) (which may be multiple since G ′ i is not connected; a minimal 2-partitionof an equilateral 3-star gives an example), such that the support of ψ i is connected. Uponreplacing G ′ i by G † i := supp ψ i and discarding G ′ i \ G † i from the support of the partition, wehave obtained an internally connected cluster whose energy is still equal to λ ( G ′ i ) = L Dk, ∞ ( G ).Repeating this process for all i = 1 , . . . , k yields an equilateral internally connected k -partitionrealising L Dk, ∞ ( G ). (cid:3) Remark 8.10.
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James B. Kennedy, Grupo de F´ısica Matem´atica, Faculdade de Ciˆencias, Universidade deLisboa, Campo Grande, Edif´ıcio C6, P-1749-016 Lisboa, Portugal
E-mail address : [email protected] Pavel Kurasov, Department of Mathematics, Stockholm University, SE-106 91 Stockholm,Sweden
E-mail address : [email protected] Corentin L´ena, Department of Mathematics, Stockholm University, SE-106 91 Stockholm,Sweden
E-mail address : [email protected] Delio Mugnolo, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, FernUniversit¨atin Hagen, D-58084 Hagen, Germany
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