Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs
IINTERLACING AND FRIEDLANDER-TYPE INEQUALITIES FORSPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS
MATTHIAS HOFMANN AND JAMES B. KENNEDY
Abstract.
We prove interlacing inequalities between spectral minimal energies of metricgraphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in[Kennedy et al , arXiv:2005.01126]. These inequalities, which involve the first Betti num-ber and the number of degree one vertices of the graph, recall both interlacing and otherinequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on thedifference between the number of nodal and Neumann domains of the whole graph eigen-functions. To this end we study carefully the principle of cutting a graph, in particularquantifying the size of a cut as a perturbation of the original graph via the notion of its rank .As a corollary we obtain an inequality between these energies and the actual Dirichlet andstandard Laplacian eigenvalues, valid for all compact graphs, which complements a versionfor tree graphs of Friedlander’s inequalities between Dirichlet and Neumann eigenvalues ofa domain. In some cases this results in better Laplacian eigenvalue estimates than thoseobtained previously via more direct methods. Introduction
The study of spectral minimal partitions on domains, first introduced in [CTV05], isby now a well-established topic; see [BBN18, BNH17, BNL17, HHT09] and the referencestherein. Roughly speaking, the goal is to find the k -partition of a domain Ω ⊂ R d , considersome eigenvalue, usually the first eigenvalue of the Dirichlet Laplacian, on each of thepartition elements, and then seek the partition which minimizes some functional of theseeigenvalues, usually their maximum. These are of interest not least because of a number ofconnections to the eigenvalues and eigenfunctions of the Dirichlet Laplacian on the wholeof the domain Ω, for example via their connection to nodal partitions, the nodal countand Pleijel’s theorem on the number of nodal domains of any k -th eigenfunction of Ω; see[BNH17, Section 10.2] for a recent survey.On compact metric graphs, the subject is newer and much less well developed. Apartfrom an earlier work studying nodal partitions in particular [BBRS12] such spectral minimalpartitions have only been systematically investigated very recently, since [KKLM20], wherewell-posedness and basic properties of a number of different spectral partitioning problemswere established. Mathematics Subject Classification.
Key words and phrases.
Metric graph; Laplacian; spectral minimal partition; spectral geometry.The authors wish to thank Gregory Berkolaiko and Delio Mugnolo for their thoughtful and helpfulcomments on an earlier version of this paper. Both authors were supported by the Funda¸c˜ao para a Ciˆenciae a Tecnologia, Portugal, via the programs “Investigador FCT”, reference IF/01461/2015 (J.B.K.) and“Bolseiro de Investiga¸c˜ao”, reference PD/BD/128412/2017 (M.H.). The work of both authors was alsopartially supported by the COST Association (Action CA18232). a r X i v : . [ m a t h . SP ] F e b M. HOFMANN AND J. B. KENNEDY
There are arguably two features which set spectral partitions of metric graphs apart fromtheir domain counterparts, both arising from the fact that metric graphs are essentiallyone-dimensional manifolds with singularities (the vertices):(1) far more general spectral functionals can be considered than on domains, includingin particular considering the smallest nontrivial eigenvalue of the Laplacian withstandard (i.e., natural, Neumann–Kirchhoff, Kirchhoff-continuity) vertex conditionsat the boundary of each of the partition elements (or clusters ) instead of Dirichlet;and(2) deciding how, and which, vertices may be cut to create the clusters also leads todifferent problems and different optima.Just as there is an intimate connection between the Dirichlet problem and the nodaldomains of Laplacian eigenfunctions on the whole object (graph, domain or manifold),the partitions involving standard Laplacians are related to the
Neumann domains of theeigenfunctions of the whole graph, as introduced and studied recently [AlBa19, ABBE20](see below).In the current context, broadly speaking, our goal is to explore the relationships betweensome of these different spectral partition problems, in particular as regards the optimalenergies (the values of the functionals at the minimizers), both in comparison with eachother and with the Laplacian eigenvalues of the whole graph. In doing so we will seestrong parallels between these energies and the way eigenvalues (and eigenfunctions) usuallybehave.In order to state our results more precisely, we first have to introduce more preciselywhich problems we will be considering; full details will be given in Section 2. Given acompact metric graph G and a k -partition P = ( G , . . . , G k ), k ≥
1, of G , that is, where the clusters G , . . . , G k are closed connected subgraphs intersecting at at most a finite number of boundary points , denote by λ ( G i ) the first eigenvalue of the Laplacian on G i with Dirichletconditions at the boundary points and standard conditions elsewhere, and by µ ( G i ) thefirst nontrivial eigenvalue of the Laplacian with standard conditions everywhere; then itwas shown in [KKLM20], among other things, that minimizing not just the functionalΛ D ( P ) := max { λ ( G ) , . . . , λ ( G k ) } but also Λ N ( P ) := max { µ ( G ) , . . . , µ ( G k ) } ( N for natural/Neumann–Kirchhoff) among suitable classes of partitions is a well-posedproblem, that is, there exists a minimizer among all P . The class of partitions of primaryinterest for us will be the most general class C k of all general connected k -partitions of G ,where we merely insist that the clusters G i be connected components of some cut graph of G ; we will also meet the more restricted class R k of rigid k -partitions of G , where we onlymake cuts at the boundary points between clusters but not in the interior of the clusters (see[KKLM20, Section 2.2] for a motivating example, and Section 2.3 below for the definitions).While somewhat technical, both these classes seem to emerge naturally in different contexts; NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 3 we will thus consider the three spectral minimal quantities L Dk, ∞ ( G ) = inf P∈ R k Λ Dk ( P ) = inf P∈ C k Λ Dk, ∞ ( P ) , L N,rk, ∞ ( G ) = inf P∈ R k Λ Nk ( P ) , L N,ck, ∞ ( G ) = inf P∈ C k Λ Nk ( P ) . It turns out that all these spectral minimal energies are in some ways good surrogates forthe eigenvalues of Laplacian operators on the whole graph – even in the “ N ” case – , andcertainly better than spectral minimal energies on domains vis-`a-vis the domain eigenvalues.For example, it was shown in [HKMP20] that all these spectral minimal energies satisfy thesame Weyl asymptotics as the eigenvalues of the Laplacian with standard vertex conditionson the graph, as well as a number of two-sided bounds strongly reminiscent of similar boundson the graph eigenvalues (as, for example, may be found in [BKKM17, Section 4]). It isthus also natural to compare these quantities, both with each other and with Laplacianeigenvalues of the whole graph.Our principal objective here is to establish sharp interlacing inequalities linking the quan-tities L Dk, ∞ and L N,ck, ∞ : here and throughout we will suppose G to be a fixed connected, com-pact, finite metric graph (again, see Section 2 for more details); β will denote the first Bettinumber of G , i.e., the number of independent cycles in the graph, and | N | the number ofvertices of G of degree 1, the leaves . Theorem 1.1.
For all k ≥ max { β, } we have (cid:16) L N,rk, ∞ ( G ) ≥ (cid:17) L N,ck, ∞ ( G ) ≥ L Dk +1 − β, ∞ ( G ) . Theorem 1.2.
For all k ≥ max { β + | N | , } we have L Dk, ∞ ( G ) ≥ L N,ck +1 − β −| N | , ∞ ( G ) . A consequence of these inequalities is that we can relate these spectral minimal energieswith the eigenvalues of the Laplacian on the whole graph, both with standard conditionsat all vertices and with Dirichlet conditions at all vertices. Indeed, denote by µ k ( G ) the k -th eigenvalue of the Laplacian with standard conditions on G (starting at µ ( G ) = 0 andcounting multiplicities) and λ k ( G , V D ) the k -th eigenvalue of the Laplacian with Dirichletconditions at a distinguished set V D of Dirichlet vertices and standard conditions on therest, which we abbreviate to λ k ( G ) := λ k ( G , V ) for when all vertices are Dirichlet vertices.Then the following result is a fairly direct consequence of Theorem 1.1. Corollary 1.3.
Let G be a (connected, compact, finite) metric graph with first Betti number β ≥ . Then for all k ≥ β + 1 we have λ k ( G ) ≥ L N,ck, ∞ ( G ) ≥ L Dk +1 − β, ∞ ( G ) ≥ µ k +1 − β ( G ) . This, and indeed the principle of interlacing inequalities between such minimal energies,have several natural motivations. For one, rather suprisingly, combining Corollary 1.3 withan upper bound on L N,ck, ∞ obtained in [HKMP20] results in the following bound which, evenas a bound on µ k , actually turns out to be better for many classes of graphs than the centralbound [BKKM17, Theorem 4.9], as we shall see below. M. HOFMANN AND J. B. KENNEDY
Corollary 1.4.
Let G be a metric graph with first Betti number β ∈ N and total length L , and suppose there exist n ≤ |E | Eulerian paths covering G , crossing at at most finitelymany points. Then for all k ≥ max { n + 1 − β, } we have (1.1) µ k ( G ) ≤ L Dk, ∞ ( G ) ≤ π L ( k + n + β − . But at a more fundamental level a key motivation for Theorems 1.1 and 1.2 arises from theeffect on graph Laplacians, of so-called surgery on the graph. Our method of proof of thesetwo theorems, which involves studying cuts of a graph and the impact this has on beingable to glue together eigenfunctions on different parts of the graph, is intimately related toboth the nodal count (and distribution of the nodal domains) of the eigenfunctions, and thenumber and distribution of the corresponding Neumann domains.Cutting a graph at a point is a simple operation which changes the topology of thegraph and which has a predictable effect on the eigenvalues of the graph, as it represents afinite rank perturbation of the associated Laplacian (see [BKKM19, Sections 3.1 and 4.1]).Cutting a graph at exactly the points x where the k -th standard Laplacian eigenfunction u k ( x ) equals 0 leads to a nodal partition , the clusters of which are the nodal domains ofthe eigenfunction. The number and distribution of these has been explored at some length;see for example [ABB18, BBRS12, Ber08]. The Neumann domains arise as the clusters ofa partition cut at the points where u (cid:48) k ( x ) = 0; the number of Neumann domains behavessimilarly as a function of k , at least in the “generic” case where (among other things) allcuts are made away from the vertices [AlBa19]. Perhaps most notably for us, it has beenshown in the generic case that the difference between the number of nodal domains ν ( k )and the number of Neumann domains ξ ( k ) of u k satisfies exactly the same bounds as theindices appearing in Theorems 1.1 and 1.2 [AlBa19, Proposition 3.1(1)] (see also [ABBE20,Proposition 11.2]):(1.2) 1 − β ≤ ν ( k ) − ξ ( k ) ≤ β + | N | − . In fact, (1.2) can also be recovered from our proofs (see Remarks 4.2 and 5.3). Despite thecompletely different approaches (here we study cutting and pasting eigenfunctions arisingfrom different minimal partitions, in [AlBa19] the point of departure being the whole grapheigenfunctions) this hints at a much deeper connection between these spectral minimalpartitions and the nodal and Neumann domain patterns of the whole graph eigenfunctions,analogous to or extending the connection between nodal domains and partitions exploredin [BBRS12], which will be left to future investigation to explore fully.Somewhat related is the idea of changing a vertex condition from standard to Dirichlet (orvice versa), another finite rank perturbation which leads to interlacing inequalities betweenDirichlet and standard Laplacian eigenvalues. A consequence of the min-max characteriza-tion of the eigenvalues is the interlacing inequality which in the notation introduced abovereads λ k + |V D | ( G , V D ) ≥ µ k + |V D | ( G ) ≥ λ k ( G , V D )(again, see [BKKM19, Section 3.1], or, e.g., [BeKu13, Section 3.1.6]). This is reminiscent of,or rather actually at odds with, Friedlander’s inequalities between Dirichlet and NeumannLaplacian eigenvalues on domains in R d , d ≥ λ k (Ω) ≥ µ k +1 (Ω) for all k ∈ N ; in fact the inequality was later shown to be always strict [Fil05].Similar results also can be obtained for compact manifolds (see [ArMa12]). On metric NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 5 graphs this is rather difficult to recover precisely because of the interlacing inequalities, or therelated idea that the difference between Dirichlet and standard vertex conditions is somehow“smaller” than the difference between Dirichlet and Neumann boundary conditions. OurCorollary 1.3 is a complement to the inequality proved in [Roh17, Theorem 4.1] for treegraphs G (i.e., with β = 0), which states that(1.3) λ k ( G ) ≥ µ k +1 ( G ) , k ∈ N , if we impose Dirichlet conditions on all vertices V D = V of G . Note, however, that (1.3)actually holds under the weaker assumption that Dirichlet conditions only be imposed on theleaves of the tree, with standard conditions at all other vertices; see [BBW15, Lemma 4.5](with t = 0). Remark 1.5.
Before proceeding, that for k sufficiently large (how large potentially depend-ing on G ), as discussed in [HKMP20] we actually have L N,ck, ∞ ( G ) = L N,rk, ∞ ( G ), as the partitionsachieving the former infimum will in fact be rigid; for such k , all our results may be ad-justed accordingly. In general, however, the class of general connected partitions seems morenatural in this context, as, unlike in the rigid case, they do not place potentially artificialrestrictions on the locations of the cuts made when forming the partitions. Hence we willnot deal with the question of rigidity further.This paper is organized as follows. We give our notation and introduce the basic notionswe will need, such of those of cuts and partitions, in Section 2. This also entails a ratherdetailed study of the notion of the cut of a graph; to this end we will introduce the rank of acut, which allows to quantify the corresponding finite rank perturbation of the correspond-ing function spaces considered. A description of said function spaces, as well as the spectralquantities we will be considering, is given in Section 3. Section 4 is devoted to the proof ofTheorem 1.1, Section 5 to the proof of Theorem 1.2; in both cases, at the beginning of thesection we include a somewhat less formal explanation of where the respective indices ap-pearing in the inequalities come from. Finally, in Section 6 we prove Corollaries 1.3 and 1.4,give several examples of graphs where the bounds in (1.1) are better than bounds obtainedelsewhere, and also study the case of certain windmill graphs , introduced in [KuSe18], wherethere is equality everywhere in (1.1).2. Metric graphs and partitions
Basic assumptions.
We start with our formalism for metric graphs. Throughoutthe article we will adopt the framework used in [KKLM20] and [Mug19], though it will benecessary to recall some of the particulars here. For us, a metric graph G = ( V , E ) will consistof a finite union E = { [ x , x ] , · · · , [ x n − , x n ] } of closed, bounded intervals in R , turnedinto a compact metric space by gluing the intervals at the endpoint set X = { x , . . . , x n } in an appropriate sense, via a partition V = { v , . . . , v m } on X , X = v (cid:116) . . . (cid:116) v m . We call each element of V , which is formally a set of endpoints, a vertex of G ; we call V the vertex set of G and E the edge set of G , whose elements, the edges, will be denoted by e ∈ E .The natural metric on G then arises by identifying each vertex with a point, treating each Thus, for us, a metric graph is what is usually called a compact metric graph in the literature.
M. HOFMANN AND J. B. KENNEDY edge e ∈ E as a subset of G , and introducing paths between pairs of points on the graph inaccordance with this identification (cf. [Mug19]). The length of an edge e , i.e., the lengthof the interval to which it corresponds, will be denoted by | e | ; the total length of G will bedenoted by L := |G| = (cid:88) e ∈E | e | . We say G is connected if it is connected as a metric space, that is, if the distance betweenany two points in the graph is finite.We can define an equivalence relation on the class of all such metric graphs via isometricisomorphisms, bijective mappings between graphs which preserve the metric; if two graphsare isometrically isomorphic to each other, then we are in one or both of the followingsituations:(i) the edge and vertex sets of one graph are permutations (i.e. relabelings) of the edgeand vertex sets, respectively, of the other;(ii) the graphs differ by the presence of dummy vertices, i.e. vertices of degree 2 that canbe added at will essentially subdividing an interval in two intervals of total length ofthe original interval.We will always identify graphs that are isometrically isomorphic, and choose a convenientrepresentative of the corresponding equivalence classes (called ur-graphs in [KKLM20]) inany given context, without further comment.2.2. Cuts of a graph.
The notion of cutting a graph will be used extensively throughoutthe paper. While it is by no means new – among other things it has appeared frequentlyin the context of spectral geometry of graphs as a prototypical “surgery principle” (see,for example, [BKKM19] and the references therein) and was also used in [KKLM20] as thebasis for defining partitions of graphs – we will need to study this notion far more carefullythan in those works, and introduce a number of new concepts around it. We thus start withthe basic definition.
Definition 2.1.
Let G and G (cid:48) be metric graphs. Then G (cid:48) is a cut (or cut graph ) of G if(i) G and G (cid:48) have a common edge set, and(ii) for all v (cid:48) ∈ V (cid:48) there exists v ∈ V such that v (cid:48) ⊂ v .We say v ∈ V is a cut vertex if there exists v (cid:48) ∈ V (cid:48) such that v (cid:48) (cid:40) v , and denote the set ofcut vertices, the cut set , by C ( G (cid:48) : G ), which we treat as a subset of G . If C ( G (cid:48) : G ) = { v } ,then we say G has been cut at v .In practice this definition allows for cutting through the interior of edges of G , as inaccordance with our observation at the end of Section 2.1 we may always insert dummyvertices at the cut locations before making the cut. The process of “undoing” a cut, i.e.,reverting to G from G (cid:48) , will as usual be called gluing. In particular, we say G has beenobtained from G (cid:48) by gluing the vertices v , . . . , v n ∈ V ( G (cid:48) ) if G (cid:48) is a cut of G such that C ( G (cid:48) : G ) = { v } , where v = v ∪ . . . ∪ v n .The next notion will be central for all our interlacing results; intuitively, it gives us directcontrol on the codimensions of the spaces of functions defined on the respective graphs. Definition 2.2.
Let G , G (cid:48) be metric graphs. Suppose G (cid:48) is a cut of G , such that the graphshave vertex sets V (cid:48) and V , respectively, then we say NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 7 (i) G (cid:48) is a cut of G of rank rank( G (cid:48) : G ) := |V (cid:48) | − |V| . (ii) G (cid:48) is a simple cut if rank( G (cid:48) : G ) = 1 , i.e. there exists a unique v ∈ V and v (cid:48) , v (cid:48) ∈ V (cid:48) such that v = v (cid:48) ∪ v (cid:48) (see also [KKLM20,Definition 2.7(1) with k = 1]). Remark 2.3.
The rank, thus defined, is invariant under relabeling of the edges and insertionor removal of dummy vertices in G (which by definition of a cut must then also be insertedor removed simultaneously in G (cid:48) ), and is hence invariant under isometric isomorphisms ofthe graph. Lemma 2.4.
Let G , G , G be metric graphs with common edge set. Suppose G is a cut of G and G is a cut of G , then(1) G is a cut of G and rank( G : G ) = rank( G : G ) + rank( G : G ); (2) if rank( G : G ) = rank( G : G ) , then G = G .Proof. Suppose G , G , G have a common edge set, and vertex sets V , V , V respectively,then (1) follows immediately from the definitions of cut and rank. Now suppose thatrank( G : G ) = rank( G : G ), then rank( G : G ) = 0 and so k := |V | = |V | . Let V = { v (1)1 , . . . , v (1) k } , V = { v (2)1 , . . . , v (2) k } . Since G is a cut of G we may assume, possibly after a relabeling, that v (1) i ⊂ v (2) i forall i = 1 , . . . , k . But since there is a bijection between the two vertex sets there must beequality, v (1) i = v (2) i for all i = 1 , . . . , k . (cid:3) Remark 2.5.
A graph G being a cut of G defines a partial ordering on the set of metricgraphs. Given a metric graph G (with a fixed vertex set, i.e., where we do not permitthe insertion or removal of dummy vertices), the set of its cut graphs becomes a partiallyordered family, and by Lemma 2.4 the rank is additive on this family. Lemma 2.6.
Let G , G (cid:48) be metric graphs. Then G (cid:48) is a cut of G of rank k ∈ N if and only ifthere exists a sequence of cuts of GG = G (0) , G (1) , · · · , G ( k − , G ( k ) = G (cid:48) such that G ( i +1) is a simple cut of G ( i ) for all i = 0 , . . . , k − .Proof. Suppose G (cid:48) is a cut of G of rank k ∈ N , where the graphs have common edge set E and vertex sets V (cid:48) , V , respectively. For the “only if” statement we give a constructive proof.Let v ∈ C ( G (cid:48) : G ) and v (cid:48) ∈ V (cid:48) such that v (cid:48) (cid:40) v , then we define V (1) = V \ { v } ∪ { v (cid:48) , v \ v (cid:48) } . Then by construction G (1) = ( V (1) , E ) is a simple cut of G and |V (1) | = |V| + 1 and G (1) is asimple cut of G . One easily sees that G (cid:48) is a cut of G (1) and by Lemma 2.4 (1) we haverank( G (cid:48) : G (1) ) = k − . M. HOFMANN AND J. B. KENNEDY
We sucessively construct metric graphs G (1) , . . . , G ( k ) such that G ( i +1) is a simple cut of G ( i ) and rank( G ( i ) : G ) = i for all i = 1 , . . . , k . Then rank( G (cid:48) : G ) = rank( G ( k ) : G ) and so byLemma 2.4 (2) we conclude that G ( k ) = G (cid:48) . The other direction is a direct consequence ofthe additivity of the rank in the sense of Lemma 2.4 (1). (cid:3) Definition 2.7.
Let G be a metric graph and V ⊂ V a distinguished vertex set. Then wecall the graph G with common edge set and vertex set V := (cid:0) V \ V (cid:1) ∪ (cid:91) v ∈V (cid:91) x ∈ v { x } the total cut (graph) of G at V . Figure 2.1.
An example of a total cut of a graph at one vertex.
Example 2.8.
Let G be the metric graph depicted in Figure 2.1. Then the total cut at theindicated vertex disconnects the graph into 3 components, and the corresponding cut is ofrank 2.We finish with the following notion, which also goes to the structure of the partial orderingof the set of all cuts of a fixed graph G . Definition 2.9.
Let G = ( V , E ) be a metric graph. Suppose G , G are cuts of G with vertexsets V and V , respectively. We define the common cut (graph) G = ( V , E ) of G , G via V = { v ∩ v ⊂ X : v ∈ V , v ∈ V such that v ∩ v (cid:54) = ∅} Equivalently, the common cut of two cut graphs G , G of G is the cut G (cid:48) of G of minimalrank such that G (cid:48) is a cut graph of G and G .2.3. Partitions.
We can now introduce the central objects of study here. The next defini-tion follows [KKLM20, Section 2].
Definition 2.10.
Let k ≥ G = ( V , E ) be a metric graph. Then:(i) P := ( G , . . . , G k ) is a connected k -partition of G , or k -partition for short, if thereexists a cut G (cid:48) such that G , . . . , G k are connected components of G (cid:48) . We refer to thecomponents G , . . . , G k as clusters ;(ii) P = ( G , . . . , G k ) is an exhaustive connected k -partition if G (cid:48) = (cid:116) ki =1 G i is a cut graphof G and G , . . . , G k are its connected components. NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 9
Since we will only be interested in connected partitions, that is, partitions whose clustersare themselves connected metric graphs, we will drop the adjective “connected” and simplyrefer to connected partitions as partitions. In principle there could be multiple cuts of G which generate P if the latter is not exhaustive, cf. Figure 2.2. However, there will alwaysbe a cut of minimal rank which gives rise to P ; we will call this cut graph the canonical cutgraph . Figure 2.2.
The clusters in Figure 2.1 are connected components of thetwo cut graphs presented. The cut graph on the left is the canonical cutgraph associated with the partition, as any other cut giving rise to thesethree clusters would have higher rank (that is, it would involve cutting theoriginal graph more), as is the case for the cut on the right. In fact, it is easyto see that the right cut graph is itself a cut of the left cut graph.
Definition 2.11.
Let k ≥ G be a metric graph and P = ( G , . . . , G k ) be a k -partitionof G = ( V , E ). Suppose G = ( V , E ) , . . . , G k = ( V k , E k ) with disjoint subsets E , . . . , E k and V , . . . , V k of E and V respectively. Then we define the canonical cut (graph) G P of G associated with the partition P as the unique cut graph of G of minimal rank suchthat G , . . . , G k are connected components of the cut graph. We will refer to the quantityrank( G P : G ) as the rank of the partition P . Remark 2.12.
Let G be a metric graph and let P be a k -partition, k ≥
1. Then, keepingthe notation of Definition 2.11, the canonical cut graph G P can be constructed as the uniquecut graph with the following properties:(i) for any cut vertex v ∈ C ( G P : G ) there exists at least one cluster G i = ( V i , E i ) and v i ∈ V i such that v i ⊂ v ;(ii) if a cut vertex v ∈ C ( G P : G ) is not divided among vertices of the G i , that is, if w := v \ (cid:83) ˜ v ∈ (cid:83) ki =1 V i is non-empty, then there is exactly one connected component of G P such that w is a vertex of that connected component.Hence the canonical cut graph G P may be described formally as the metric graph with thesame edge set as G and vertex set V P = k (cid:91) i =1 V i ∪ v \ (cid:91) ˜ v ∈ (cid:83) ki =1 V i ˜ v : v ∈ V \ k (cid:91) i =1 V i . Definition 2.13.
Let G be a metric graph and let P = {G , . . . , G k } be a (non-exhaustive) k -partition of G , k ≥
1, with edge sets E , . . . , E k ⊂ E . We say that P (cid:48) = {G (cid:48) , . . . , G (cid:48) k } is an exhaustive extension of P if(i) G P is a cut of G P (cid:48) (ii) E i ⊂ E (cid:48) i for all i = 1 , . . . , k (iii) (cid:83) ki =1 E (cid:48) i = E .We next introduce the following notation for the boundary points of a partitions. Definition 2.14.
Let P = ( G , . . . , G k ) be a (not necessarily exhaustive) k -partition of G , k ≥
1, and let G P the canonical cut graph of G associated with P .(i) We say G i = ( V i , E i ) and G j = ( V j , E j ), i (cid:54) = j , are neighboring clusters , or just neighbors ,if there exist v i ∈ V i , v j ∈ V j and v ∈ V such that v i , v j ⊂ v . We call any such v ∈ V a boundary vertex (or boundary point ) of P , and define the boundary set of P to be theset of all such boundary points: ∂ P := { v ∈ V : ∃ v i ∈ V i , v j ∈ V j , i (cid:54) = j : v i , v j ⊂ v } . (ii) We define the boundary set of the cluster G i = ( V i , E i ) by ∂ G i = { v ∈ V i : ∃ v (cid:48) ∈ C ( G P : G ) : v (cid:40) v (cid:48) } (cf. Definition 2.1).For the sake of completeness, we also recall the definition of rigid partitions from [KKLM20],where cuts can only be made at the boundary between neighbors; these can be characterizedconveniently using the notion of canonical cut graphs. Definition 2.15.
We say a (connected) k -partiton P = ( G , . . . , G k ) of G is rigid if itsboundary set ∂ P coincides with the cut set C ( G P : G ).We denote the class of all (connected) exhaustive k -partitions of G by C k ( G ) and theclass of all rigid exhaustive k -partitions of G by R k ( G ). We finish this section with a usefulestimate. Lemma 2.16.
Let G be a metric graph with first Betti number β ≥ . Suppose P =( G , . . . , G k ) is an exhaustive k -partition of G , k ≥ . Then (2.1) k − ≤ rank( G P : G ) ≤ k − β. Proof.
Assume without loss of generality that G = ( V , E ) and G = ( V , E ) are neighbor-ing clusters. Suppose there exists v ∈ V , v ∈ V and v ∈ V such that v , v ⊂ v . Weglue G and G at v , i.e. we obtain a graph G (1) with edge set E (1) = E ∪ E and vertexset V (1) = V ∪ V ∪ { v ∪ v } \ { v , v } . By construction P (1) = {G (1) , G , . . . , G k } definesa k − G P is a simple cut of G P (1) . Applying this procedure iteratively andinvoking Lemma 2.6, we end up with an exhaustive 1-partition G ( k − such that G P is a cutof G ( k − , G ( k − is a cut of G , and rank( G P : G ( k − ) = k −
1. Lemma 2.4 now yields thelower bound in (2.1).On the other hand, since G admits β independent cycles any cut of rank β + 1 wouldnecessarily disconnect G ; since G ( k − is a connected cut graph of G we thus have rank( G ( k − : G ) ≤ β . Lemma 2.4 now yields the upper bound in (2.1). (cid:3) NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 11 Function spaces and spectral functionals
Given a metric graph G = ( V , E ) the spaces C ( G ) , L ( G ) of continuous and square inte-grable functions, and the Sobolev space H ( G ), respectively, may be defined in the usualway, cf. [Mug19]: C ( G ) = (cid:40) f ∈ (cid:77) e ∈E C ( e ) : f ( x ) = f ( y ) for all x, y ∈ v, for all v ∈ V (cid:41) ,L ( G ) = (cid:77) e ∈E L ( e ) , H ( G ) = C ( G ) ∩ (cid:77) e ∈E H ( e ) . Given a distinguished set of vertices V D (which may include dummy vertices) we also define H ( G , V D ) = { f ∈ H ( G ) : f ( v ) = 0 for all v ∈ V D } . If G (cid:48) is a cluster of a partition of G and f ∈ H ( G (cid:48) , ∂ G (cid:48) ), then we will identify f with afunction in H ( G ) in the canonical way, by extension to zero outside G (cid:48) . We also record thefollowing notions related to the zero set of an H -function for later use. Definition 3.1.
Given f ∈ H ( G ),(i) its nodal set is defined to be N ( f ) = { x ∈ G : f ( x ) = 0 } , and we call x ∈ N ( f ) a nodal point (of f );(ii) we call each connected component of G \ N ( f ) a nodal domain of f ;(iii) we say a nodal point x ∈ N ( f ) is non-degenerate if f (cid:54)≡ x .Whenever f is a Laplacian eigenfunction (see below) its set of non-degenerate nodalpoints is finite; thus we may assume without loss of generality that they are vertices. A nodal partition is the non-exhaustive partition of G whose clusters are the (closures of the)nodal domains of f . More precisely, its clusters are those connected components on which f does not vanish identically, of the total cut graph of G at all non-degenerate nodal points.We will be interested in the smallest nontivial eigenvalue of the Laplacian subject toDirichlet conditions on V D and standard, a.k.a. natural (continuity plus Kirchhoff) condi-tions at all other vertices. If V D = ∅ this eigenvalue may be described variationally by(3.1) µ ( G ) = inf (cid:40) ´ G | f (cid:48) ( x ) | d x ´ G | f ( x ) | d x : f ∈ H ( G ) \ { } such that ˆ G f d x = 0 (cid:41) ;we will also, very loosely, refer to this problem as the “Neumann case”, or N case. If at leastone vertex is equipped with Dirichlet conditions, then the smallest nontrivial eigenvalue is(3.2) λ ( G , V D ) = inf (cid:40) ´ G | f (cid:48) ( x ) | d x ´ G | f ( x ) | d x : f ∈ H ( G , V D ) \ { } (cid:41) and we talk of the Dirichlet case. Here, if the Dirichlet vertex set V D is clear we will justwrite λ ( G ). In both cases equality is attained exactly at the corresponding eigenfunctions.As usual we will call the ratio appearing in (3.1) and (3.2) the Rayleigh quotient of f . As in [KKLM20], given a k -partition P = ( G , . . . , G k ) of G we consider the spectralenergies Λ Nk ( P ) := max { µ ( G ) , . . . , µ ( G k ) } , Λ Dk ( P ) := max { λ ( G ) , . . . , λ ( G k ) } , where unless explicitly stated otherwise we will always take the Dirichlet eigenvalue λ ( G i )with zero set V D = ∂ G i being the boundary set of the cluster. The subscript k will al-ways denote the number of clusters of the partition; if for a given partition this number isunknown, then we will omit it and simply write Λ N ( P ), Λ D ( P ). The k -partitions minimizing the energies Λ Nk , Λ Dk among all (suitable) k -partitions arecalled spectral minimal k -partitions ; the concrete minimization problems we will considerhere are to find L N,rk, ∞ ( G ) = inf P∈ R k Λ Nk ( P ) , L N,ck, ∞ ( G ) = inf P∈ C k Λ Nk ( P ) L Dk, ∞ ( G ) = inf P∈ R k Λ Dk ( P ) = inf P∈ C k Λ Dk ( P ) . In all cases the results of [KKLM20, Section 4] guarantee the existence of k -partitionsattaining these respective infima; there it is also shown that there is equality between thetwo classes in the Dirichlet case. We need to extend these results slightly, to ensure that theinfimum is the same regardless of whether we minimize over exhaustive or not necessarilyexhaustive k -partitions. Lemma 3.2.
Let G be a metric graph and let P ∈ C k ( G ) be a k -partition of G , k ≥ . Thenthere exists an exhaustive extension P (cid:48) ∈ C k ( G ) such that Λ Nk ( P ) ≥ Λ Nk ( P (cid:48) ) and Λ Dk ( P ) ≥ Λ Dk ( P (cid:48) ) . Proof.
We deal with the N case; in the Dirichlet case the argument is the same. Suppose P = ( G , . . . , G k ) and let u , . . . , u k be any eigenfunctions associated with µ ( G ) , . . . , µ ( G k ),respectively. Suppose the canonical cut graph G P = ( V P , E ) associated with P has connectedcomponents G , . . . , G k , G k +1 , . . . , G m , where we assume that m > k . Let G i = ( V i , E i ) for all i = 1 , . . . , m .Let v ∈ ∂ P be a boundary vertex between some G i , i ≤ k , and G j , k < j ≤ m , sothat there exist v i ∈ V i for some i ≤ k and v j ∈ V j for some k < j ≤ m such that v i , v j ⊂ v . We define a (first) extension P (1) = ( G , . . . , G i − , G (1) i , G i +1 , . . . , G k ), where G (1) i is the (connected) union of G i and G j , glued at v i and v j : in particular, the canonical cutgraph G P (1) has edge set E and vertex set V P (1) = V P \ { v i , v j } ∪ { v i ∪ v j } ;Since G i and G j are glued at a single vertex of G (1) i , [BKKM19, Theorem 3.4] yields µ ( G (1) i ) ≤ µ ( G i ) and hence Λ Nk ( P (1) ) ≤ Λ Nk ( P ). We successively construct partitions P ( (cid:96) +1) from Note that here our notation is slightly different from that in [KKLM20], as here the subscript gives thenumber of clusters k rather than the value of p in the p -norm of the combination of eigenvalues. The reasonis that here it will be important to keep track of the number of clusters of our partitions, whereas we alwaysconsider functionals built via the ∞ -norm of the vector of eigenvalues. NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 13 P ( (cid:96) ) such that Λ Nk ( P ( (cid:96) +1) ) ≤ Λ Nk ( P ( (cid:96) ) ). Then P (cid:48) := P ( m − k ) is necessarily exhaustive, andΛ Nk ( P (cid:48) ) ≤ Λ Nk ( P ). (cid:3) Corollary 3.3.
For all k ≥ we have L N,ck, ∞ ( G ) = inf { Λ Nk ( P ) : P is a (not necessarily exhaustive) k -partition of G} , L Dk, ∞ ( G ) = inf { Λ Dk ( P ) : P is a (not necessarily exhaustive) k -partition of G} . That is, when minimizing the energies Λ Nk , Λ Dk among all connected k -partitions, it makesno difference whether we minimize over all k -partitions, or over all exhaustive k -partitions.4. Proof of Theorem 1.1
Assume G is a connected metric graph with first Betti number β = |E | − |V| + 1. Wefirst wish to give an intuitive explanation as to why Theorem 1.1 should hold. So let P = ( G , . . . , G k ) ∈ C k ( G ) be an exhaustive k -partition realizing the minimum for L N,ck, ∞ ( G ).Consider any eigenfunctions u , . . . , u k on G , . . . , G k associated with µ ( G ) , . . . , µ ( G k ), re-spectively. We extend each function by zero to obtain an L -function on G , which can alsobe treated as an element of (cid:76) H ( e ), and which we still denote by u i , i = 1 , . . . , k . Nowsince each of these functions necessarily changes sign the sets u i, + = { u i > } u i , u i, − = { u i < } u i ,i = 1 , . . . , k , are all non-empty. Suppose we can match these eigenfunctions at the cutvertices in the sense that there exist α , + , α , − , . . . , α k, + , α k, − ∈ R \ { } such that for all v ∈ C ( G P : G )(4.1) α i, sign( u i ( v )) u i ( v ) = α j, sign( u j ( v )) u j ( v )for all v , v ∈ C v ( G P ). Then(4.2) u := α , + u , + + α , − u , − + . . . + α k, + u k, + + α k, − u k, − ∈ H ( G ) . How many nodal domains will u have on G ? We know that:(1) regarded as a function on the cut graph G P it has at least 2 k , since it changes signon each connected component G i , i = 1 , . . . , k , of G P ;(2) by Lemma 2.16 we have k − ≤ rank( G P : G ) ≤ k − β ;(3) every time we make a cut of G of rank 1 (cf. Lemma 2.6) the number of nodaldomains of u considered as a function on the cut graph increases by at most 1.It follows that u ∈ H ( G ) admits at least 2 k − rank( G P : G ) ≥ k + 1 − β nodal domains;moreover, its Rayleigh quotient on each of these nodal domains will be no larger thanΛ Nk ( P ) = max i µ ( G i ). Thus, if u ∈ H ( G ), then we can use the associated nodal partitionto obtain L N,rk, ∞ ( G ) ≥ L N,ck, ∞ ( G ) ≥ L D k − rank( G P : G ) , ∞ ( G ) ≥ L Dk +1 − β, ∞ ( G ) , which is Theorem 1.1. But of course in general we cannot expect the matching conditions(4.1) to hold. Example 4.1.
Before proceeding we give a simple example to show that equality is possiblein both Theorem 1.1, that is, that we may have equality(4.3) L N,rk, ∞ ( G ) = L Dk +1 − β, ∞ ( G ) , as well as in the above argument. Consider the equilateral m -star, m ≥
3, with m edges oflength 1 each. We identify each edge e with the unit interval [0 , L N,rjm, ∞ ( S m ) = π j = L Djm +1 , ∞ ( S m ) . for all j ≥
1. Moreover, in this case there is a nodal partition corresponding to L Djm +1 , ∞ ( S m ),which comes from taking eigenfunctions of the form u e,j ( x ) = cos( πjx ) on each edge e (cid:39) [0 , k not of the form jm + 1, since forexample L N,rjm − , ∞ = π m j L > π m ( j − / L = L Djm, ∞ for all j ≥ k ≥ Remark 4.2.
Suppose u is an eigenfunction, with eigenvalue λ , of the (standard) Laplacianon G , and suppose that considering the total cut of G at all points where u reaches a localnonzero maximum or minimum generates a partition with k = ξ ( u ) clusters. (In the lan-guage of Section 5 and [ABBE20] this means u has ξ ( u ) Neumann domains .) Then λ equalsthe first nontrivial standard Laplacian eigenvalue on each cluster, with eigenfunction u (see[ABBE20, Lemma 8.1]). Now by construction we can certainly match these restrictions ofthe eigenfunction at the cut points, in accordance with the above discussion. As we haveseen, the resulting Dirichlet partition consists of at least ξ ( u ) + 1 − β clusters, which in thiscase are clearly the nodal domains of u . Thus we recover one part of (1.2). Lemma 4.3.
Suppose G (cid:48) is a simple cut of G . Suppose ˜ u ∈ H ( G (cid:48) ) with nodal partition P = ( G , . . . , G n ) with n ∈ N . Then there exists a function u ∈ span(˜ u (cid:12)(cid:12) G , . . . , ˜ u (cid:12)(cid:12) G n ) ∩ H ( G ) with at least n − nodal domains.Proof. Suppose that v is the unique vertex in C ( G (cid:48) : G ), and v , v ∈ V (cid:48) such that v = v ∪ v .Suppose without loss of generality that v ∈ G and v ∈ G . Write G i = ( V i , E i ), i = 1 , . . . , n . Case 1: ˜ u ( v ) = 0 and ˜ u ( v ) = 0 . Then since ˜ u ( v ) = ˜ u ( v ) we infer ˜ u ∈ H ( G ) and weare done since ˜ u admits n connected nodal domains. Case 2: ˜ u ( v ) (cid:54) = 0 (cid:54) = ˜ u ( v ) . Then there exist α , α (cid:54) = 0 such that α ˜ u ( v ) = α ˜ u ( v )and we may define u ( x ) := α ˜ u ( x ) , x ∈ E α ˜ u ( x ) , x ∈ E ˜ u ( x ) otherwise,so that u ∈ H ( G ) with n − NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 15
Case 3: Otherwise.
Suppose without loss of generality that ˜ u ( v ) (cid:54) = 0 and ˜ u ( v ) = 0.Then we define u ( x ) := (cid:40) ˜ u ( x ) , x (cid:54)∈ G , otherwise,and by construction u ∈ H ( G ) has n − (cid:3) Lemma 4.4.
Let G be a metric graph and G (cid:48) a cut of G . Let r := rank( G (cid:48) : G ) and k > r , then for any k -partition P (cid:48) = ( G (cid:48) , . . . , G (cid:48) k ) of G (cid:48) there exists a ( k − r ) -partition P = ( G , . . . , G k − r ) of G such that Λ Dk ( P (cid:48) ) ≥ Λ Dk − r ( P ) . Proof.
By a simple induction argument based on Lemma 2.6 it suffices to prove the resultfor r = 1. So suppose G (cid:48) is a simple cut of G and P (cid:48) = ( G (cid:48) , . . . , G (cid:48) k ) is an arbitrary k -partitionof G (cid:48) . We let ˜ u i ∈ H ( G (cid:48) i , ∂ G (cid:48) i ) be an eigenfunction associated with λ ( G (cid:48) i ), i = 1 , . . . , k , thenthe function ˜ u such that ˜ u | G (cid:48) i = ˜ u i for all i belongs to H ( G (cid:48) ) and has nodal partition exactly P (cid:48) , and . Then by Lemma 4.3 there exists u ∈ H ( G ) with at least k − u | G (cid:48) i = ˜ u i for all i . A simple argument using the nodal partition P associatedwith u and fact that Λ Dk ( P (cid:48) ) = max i λ ( G (cid:48) i ) leads to Λ Dk ( P (cid:48) ) ≥ Λ Dk − r ( P ). (cid:3) Corollary 4.5.
Let G be a metric graph and G (cid:48) a cut of G , and suppose r := rank( G (cid:48) : G ) .Then L Dk, ∞ ( G (cid:48) ) ≥ L Dk − r, ∞ ( G ) . Lemma 4.6.
Let G be a metric graph and let P = ( G , . . . , G k ) be a k -partition with canonicalcut graph G P . Let r := rank( G P : G ) , then there exists a (2 k − r ) -partition P (cid:48) = ( G , . . . , G k − r ) such that Λ Nk ( P ) ≥ Λ D k − r ( P (cid:48) ) . Proof.
Suppose G P is the canonical cut graph of P = ( G , . . . , G k ). Let u i be an eigenfunctionfor µ ( G i ) on G i , i = 1 , . . . , k . Then u i necessarily changes sign on G i and hence admitsat least two nodal domains; denote by P i = ( G i, + , G i, − ) any exhaustive extension of anycorresponding nodal 2-partition of G i . Then, since µ ( G i ) ≥ max { λ ( G i, + ) , λ ( G i, − ) } ,Λ Nk ( P ) ≥ max i =1 ,...,k max { λ ( G i, + ) , λ ( G i, + ) } ≥ L D k, ∞ ( G P ) ≥ L D k − r, ∞ ( G ) , where the last inequality follows from Corollary 4.5. (cid:3) We can now give the proof of Theorem 1.1.
Proof of Theorem 1.1.
Let P be any k -partition of G . By Lemma 2.16 we haverank( G P : G ) ≤ k − β and so, applying Lemma 4.6, taking the infimum over all such partitions and using themonotonicity of the mapping j (cid:55)→ L Dj, ∞ ( G ), we obtain L N,ck, ∞ ( G ) ≥ L Dk +1 − β, ∞ ( G ) . (cid:3) Proof of Theorem 1.2
Just as the basic idea behind Theorem 1.1 is gluing together nodal domains of Neumanneigenfunctions of the partition clusters to construct a test partition, here we will be interestedin gluing together the so-called
Neumann domains of the cluster Dirichlet eigenfunctions(see, e.g., [AlBa19, ABBE20]).Let us again start with an intuitive explanation of Theorem 1.2. We suppose G is a metricgraph with first Betti number β and | N | leaves. We take P = ( G , . . . , G k ) ∈ C k ( G ) to bea fixed exhaustive k -partition of G and consider the respective first Dirichlet eigenfunctions u , . . . , u k on G , . . . , G k , associated with λ ( G ) , . . . , λ ( G k ) and extended by zero on the restof G .We decompose each G i by taking the total cut (see Definition 2.7) of G i at every point,without loss of generality a vertex v ∈ V ( G i ), at which u i attains a nonzero extremum,and thus in particular ∂∂ν | e u i ( v ) = 0 on every edge e incident with v . On each connectedcomponent (cid:101) G i, , . . . , (cid:101) G i,k i , k i ≥
1, the Neumann domains, u i is the first eigenfunction of theLaplacian with suitable mixed Dirichlet-natural conditions, and in particular λ ( G i ) is stillthe first eigenvalue of each (cid:101) G i,j by a standard variational argument (cf. [BKKM17, Proof ofTheorem 3.4], or also [ABBE20, Lemma 8.1] for a similar principle).Now suppose that, given a cut vertex v ∈ C ( G P : G ), we glue together all the neighboringNeumann domains G i ,j , . . . , G i kv ,j kv at v to form a cluster G (cid:48) := G i ,j ∪ . . . ∪G i kv ,j kv ; then, bytaking a suitable linear combination of u i | G i ,j , . . . , u i kv | G ikv ,jkv similarly to (4.2), we obtaina test function on G (cid:48) , orthogonal to the constant functions for the right choice of coefficients,whose Rayleigh quotient is at most max { λ ( G i ) , . . . , λ ( G i kv ) } ≤ Λ Dk ( P ).Gluing such neighboring Neumann domains together at as many different cut verticesas possible (see also Figure 5.1), we may thus construct a partition P (cid:48) of G such thatΛ N ( P (cid:48) ) ≤ Λ Dk ( P ).The question is, how many clusters can P (cid:48) have? Denote by (cid:101) P = { (cid:101) G i,j } i,j the partitionof G which results from making total cuts of the clusters of P as described above, whichwill be a finer partition than P and P (cid:48) (in fact, G (cid:101) P will be the common cut of G P and G P (cid:48) ,cf. Definition 2.9). We wish to determine how many clusters must be created when passingfrom P to (cid:101) P , and how many may be lost from (cid:101) P to P (cid:48) .For the first question, we wish to find a condition that guarantees that a cluster G i of P will yield (at least) two in (cid:101) P , that is, that it contains at least two Neumann domains. Asufficient condition is that G i have at least two Dirichlet (cut) vertices, and that u i reach anextremum on every trail (non-self-intersecting path) in G i connecting them. Observe thatthis need not be the case if the cluster contains a leaf or a cycle of G (for example if G i is aninterval with one Dirichlet and one Neumann condition, or lasso with a Dirichlet conditionat its degree-one vertex). This motivates the following definition. Definition 5.1.
Suppose P = ( G , . . . , G k ) is a k -partition, k ≥
2, of G . We say that acluster G i is benign (in G ) if it contains neither a vertex of G of degree one, nor a cycle of G ; otherwise, we say it is malign .Observe that any benign cluster of G must necessarily be a tree each of whose leavesbelongs to the cut set C ( G P : G ) = ∂ P , while for malign clusters this is not necessarily thecase. We see that if P has k (cid:48) malign clusters, then (cid:101) P must have at least 2 k − k (cid:48) . NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 17
The next question is how many clusters we may lose going from (cid:101) P to P (cid:48) ; the example ofFigure 5.1 shows that the answer may be complicated. · · ·· · · Figure 5.1.
A possible cluster of P (cid:48) resulting by gluing the correspondingNeumann domains at the cut vertices of P , which are the Dirichlet points(open circles) of the associated eigenfunctions. Observe that while this clusteris composed of a large number of Neumann domains, being constructed in thisway it necessarily contains in its interior at least one boundary point of theoriginal Dirichlet partition.The following lemma formalizes the above reasoning and answers the latter question; theproof of Theorem 1.2 will then follow easily. Lemma 5.2.
Let G be a compact, connected metric graph. Suppose P = ( G , . . . , G k ) is anexhaustive k -partition, k ≥ , with rank( G P : G ) = k − r for some ≤ r ≤ β and suppose that P contains at most ≤ n ≤ k malign clusters. Thenthere exists an exhaustive k + 1 − n − r -partition P (cid:48) of G such that Λ Dk ( P ) ≥ Λ Nk +1 − n − r ( P (cid:48) ) . Proof.
Let u , . . . , u k be the respective first Dirichlet eigenfunctions on G , . . . , G k , associatedwith λ ( G ) , . . . , λ ( G k ), identified as functions in H ( G ) via extension by zero. Take P (cid:48) tobe the partition of G associated with the cut G P (cid:48) of G consisting of the total cut of G atall points where any of the u i admit a local nonzero extremum. Let (cid:101) P be the partition of G whose clusters are the connected components by the common cut graph of G P and G P (cid:48) .Then by construction rank( G P (cid:48) : G ) = rank( G (cid:101) P : G ) − rank( G P : G ) . It follows from Lemma 2.4 thatrank( G (cid:101) P : G P (cid:48) ) = rank( G (cid:101) P : G ) − rank( G P (cid:48) : G ) = rank( G P : G ) , that is, rank( G (cid:101) P : G P (cid:48) ) = k − r .Next observe that every benign cluster admits at least two Neumann domains and there-fore (cid:101) P has at least 2 k − n clusters. Lemma 2.6 combined with a simple induction argumentshows that P (cid:48) has at least (2 k − n ) − ( k − r ) = k + 1 − n − r clusters, since undoinga simple cut (i.e., gluing once) will change the number of connected components of the cutgraph by at most one. Figure 5.2.
On the left is an example of a possible cluster of P with nodes(open circles) and local extrema (crosses) of the eigenfunction. Such an eigen-function may have multiple local extrema within the cluster. However, thelocal extrema cannot enclose an area as in the image on the right. In partic-ular, any cluster of (cid:101) P , and thus of P (cid:48) , necessarily contains a node, that is, aboundary vertex of P .We claim that every cluster G (cid:48) of P (cid:48) satisfies µ ( G (cid:48) ) ≤ Λ Dk ( P ), which will complete theproof of the lemma. To this end, fix such a cluster G (cid:48) of P (cid:48) ; we first observe that G (cid:48) containsat least two clusters of (cid:101) P , that is, it is formed out of at least two distinct Neumann domainsof the eigenfunctions u , . . . , u k (cf. also Figure 5.1). To see this, observe that:(1) the boundary sets of P and P (cid:48) are disjoint: at any cut vertex of G P all the u i ∈ H ( G )satisfy a Dirichlet condition; hence no such point can also be a local nonzero extremum;(2) by construction, on each cluster of (cid:101) P there is exactly one eigenfunction u i which doesnot vanish identically, and this eigenfunction does not change sign within the cluster;(3) no eigenfunction has a strictly positive local minimum or strictly negative local maxi-mum anywhere; hence no eigenfunction can have a Neumann domain strictly containedin a nodal domain (see also Figure 5.2).If G (cid:48) should coincide with a single cluster (cid:101) G of (cid:101) P , then in particular the two must sharea boundary set. This means, by construction of (cid:101) P and (1), that (cid:101) G contains no Dirichletpoints, that is, the only eigenfunction u i (from (2)) which does not vanish identically in (cid:101) G ,cannot have any zeros whatsoever there. But this is a contradiction to (3).As a result, we can guarantee the existence of a function ϕ ∈ H ( G (cid:48) ) such that ˆ G (cid:48) ϕ ( x ) d x = 0 , by taking ϕ to be a suitable linear combination of the restrictions of u i | G (cid:48) , i = 1 , . . . , k . Then ϕ is a valid test function for µ ( G (cid:48) ) on the one hand, and on the other the Rayleigh quotientof ϕ cannot exceed Λ Dk ( P ) = max i =1 ,...,k λ ( G i ). The latter claim follows from a standardargument: by construction, on every nodal domain of ϕ we have that ϕ is a multiple ofsome u i , and thus its Rayleigh quotient is no larger than the maximum of the Rayleighquotients of the u i on the respective nodal domains. Moreover, u i satisfies either a standardor a Dirichlet condition at every vertex of this nodal domain, treated as a subgraph of G (cid:48) ,and is thus a non-sign-changing classical eigenfunction there, so, as noted earlier, λ ( G i ) isequal to the Rayleigh quotient of u i on Ω. (cid:3) NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 19
Proof of Theorem 1.2.
The theorem will follow immediately from Lemma 5.2 once we haveshown that any exhaustive partition P of G of rank k − r , 0 ≤ r ≤ β , can have at most n = β + | N | − r malign clusters.We first observe that at most | N | clusters can contain at least one leaf of G ; it remainsto show that at most β − r clusters can contain a cycle of G . But this follows if we canshow that the (disconnected) canonical cut graph G P has Betti number β − r . This, inturn, follows from a simple induction argument using the definition of r and Lemmata 2.6and 2.16: there will exist an intermediate cut of G rank r which remains connected andhas Betti number β − r ; G P is then obtained from this intermediate graph by cutting k − P ) from the rest of the graph. (cid:3) Remark 5.3.
Let u be an eigenfunction of the Laplacian on G and P be its nodal partitionwith k = ν ( u ) nodal domains (clusters). We know that on each the restriction of u to eachcluster coincides with the corresponding first eigenfunction on that cluster, with Dirichletconditions at the boundary points. Then by construction, the partition P (cid:48) in Lemma 5.2coincides with the partition of G into the Neumann domains of u . The proof of Theorem 1.2in particular ensures that this partition contains at least(5.1) ξ ( u ) ≥ ν ( u ) + 1 − β − | N | clusters, the Neumann domains of u . Combining (5.1) and Remark 4.2, we recover (1.2).6. Application: Spectral inequalities
In this section we will prove Corollary 1.3, relating the interlacing inequalities of Theo-rems 1.1 and 1.2 to the eigenvalues of the Laplacian on the whole graph G with Dirichletand standard vertex conditions. Afterwards, we will discuss their relation with concreteestimates on the optimal energies L Dk, ∞ ( G ), L N,rk, ∞ ( G ) in terms of geometric and topologi-cal properties of G ; complementary estimates were obtained in [HKMP20, Theorems 3.1and 3.2]. We recall that λ k ( G , V ) =: λ k ( G ) and µ k ( G ) are, respectively, the k -th eigenvalue,counted with multiplicities, of the Laplacian with Dirichlet conditions at all vertices of G (which thus reduces to a disjoint union of n intervals), and of the Laplacian with standardconditions at all vertices of G . Proof of Corollary 1.3.
We clearly only have to prove the first and the last inequalities, themiddle one being contained in Theorem 1.1. For the first inequality, L N,ck, ∞ ( G ) ≤ λ k ( G ), weobserve, firstly, that for any finite interval I ⊂ R and j ∈ N , λ j ( I ) = µ j +1 ( I ).We suppose that for each i = 1 , . . . , n , j i := max { j ≥ λ j ( e i ) ≤ λ k ( G ) } , so that the collection { λ (cid:96) ( e i ) : 1 ≤ (cid:96) ≤ j i } gives exactly the first k eigenvalues λ ( G ) , . . . , λ k ( G ),counted with multiplicities (if λ k ( G ) is multiple, meaning at least two edges have the sameeigenvalue corresponding to λ k ( G ), then we arbitrarily choose a certain number to be ex-cluded in order to guarantee that { λ (cid:96) ( e i ) : 1 ≤ (cid:96) ≤ j i } does in fact consist of exactly k elements, the largest of which is λ k ( G )).For each i = 1 , . . . , n for which j i ≥
1, we partition the edge e i into j i equal subintervals e i, , . . . , e i,j i , each of which is a nodal domain for the eigenfunctions of λ j i ( e i ), so that, with our first observation, µ ( e i, ) = . . . = µ ( e i,j i ) = λ ( e i, ) = λ j i ( e i ). Since (cid:80) ni =1 j i = k , the(non-exhaustive) partition P := { e i,(cid:96) : 1 ≤ (cid:96) ≤ j i , ≤ i ≤ n } is a k -partition of G such thatΛ Nk ( P ) = max i,(cid:96) µ ( e i,(cid:96) ) = max i λ j i ( e i ) = λ k ( G ) . The inequality now follows from Lemma 3.2. The last inequality, µ k ( G ) ≤ L Dk, ∞ ( G ) , follows from a standard argument involving the min-max characterization of µ k ( G ), see also[KKLM20, Proposition 8.5] for a detailed proof. (cid:3) We now turn to Corollary 1.4. We first recall that [HKMP20] derived, among other things,both upper and lower bounds for the quantities L Dk, ∞ ( G ) and L N,ck, ∞ ( G ), namely(6.1) π L k ≤ L N,ck, ∞ ( G ) ≤ π L (cid:0) k + n − (cid:1) and(6.2) π kL (cid:0) k + 3( k − | P | − | N | ) (cid:1) ≤ L Dk, ∞ ( G ) ≤ π L (cid:18) k + (cid:18) |E | − − (cid:22) | N | (cid:23)(cid:19)(cid:19) for all sufficiently large k depending on G ; see [HKMP20, Theorems 4.5, 4.9, 5.1 and 5.3](note that the proof of the upper bound in (6.1), given for L N,rk,p ( G ), works for L N,ck,p ( G )whenever k ≥ n , that is, (6.1) is valid for all k ≥ n ). Here | P | ≤ β is the number of doublyconnected pendants of G and n ≤ |E | is any number for which there exists an n -partition of G each of whose clusters consists of a single Eulerian path . Proof of Corollary 1.4.
This is an immediate consequence of the upper bound in (6.1) andCorollary 1.3. (cid:3)
We observe that our inequality (1.1), which we reproduce here for the sake of convenience, µ k ( G ) ≤ L Dk, ∞ ( G ) ≤ π L ( k + n + β − involves rather different quantities from the upper bound in (6.2), as well as what is possiblythe best general upper bound on µ k ( G ) to date, namely [BKKM17, Theorem 4.9](6.3) µ k ( G ) ≤ π L ( k + β + | N | − for all k ≥ L Dk, ∞ ( G ) ≥ π L ( k − − β − | N | ) , will not in general be better than the lower bound in (6.2), at least for large k , as onecan see by comparing the respective coefficients of the k term in the bounds. It would beinteresting to understand what the optimal coefficients might look like. NTERLACING INEQUALITIES FOR SPECTRAL MINIMAL PARTITIONS 21
Figure 6.1.
A graph given by two 3-pumpkins connected by an edge. Thegraph admits an Eulerian path seen on the right side.
Example 6.1.
We consider the pumpkin dumbbell depicted in Figure 6.1, consisting of two3-pumpkins connected by an edge (interestingly, the relative edge lengths are irrelevant forthese bounds). Then by Corollary 1.4 we have L Dk, ∞ ( G ) ≤ π L ( k +4) for all k ≥
1, while since |E | = 7 and | N | = 0 the upper bound in (6.2) reads L Dk, ∞ ( G ) ≤ π L ( k + 7) (for sufficientlylarge k ). Introducing thicker pumpkins would lead to the same conclusion, that (1.1) isbetter. . . .. . . Figure 6.2.
Stower graphs as an example of a class of graphs for which (1.1)is better than (6.3) .
Example 6.2.
The bound on µ k ( G ) in (1.1) is better than (6.3) for all flower graphs (where | N | = 0 and n = 1), and more generally stower graphs (flowers with a finite numberof pendant edges attached to the central vertex, i.e., a union of a flower and a star; seeFigure 6.2). These were introduced in [BaL´e17], where they played a major role in theminimization of µ ( G ) among various classes of graphs. There exist such stower graphs withany β ≥ | N | ≥ n ≤ (cid:100) | N | (cid:101) , leading to theassertion that (1.1) is better. Finally, the respective upper bounds coincide for star graphsfor which | N | is even, since then β = 0 and n = | N | . Example 6.3.
Let G be a windmill graph W m , m ≥
1, which consists of 2 m lassos ( blades )glued together at a central vertex (see Figure 6.3); we assume that all the loops have acommon length (cid:96) > s >
0. It was shown in [KuSe18]that, if the ratio (cid:96)/s = 4, then there is equality in (6.3) for any number of blades. Inparticular, since β = 2 m , µ k ( W m ) = π L ( k + 3 m − for all k ≥
1. Note that W m can be partitioned into n = m clusters, each consisting ofexactly two blades glued together (like the dumbbell pumpkin of Figure 6.1 but with loops ... ... · · · Figure 6.3.
Windmill graphs are examples of graphs for which the upperestimate in (6.3) is attained. As it turns out this is also the case in (1.1) whenthe graph has an even number of pendant lassos.in place of the 3-pumpkins). This means that the upper bound in (1.1) is also equal to π L ( k + 3 m − ; hence we have equality everywhere, µ k ( W m ) = L Dk, ∞ ( W m ) = L N,ck − β, ∞ ( W m ) = π L ( k + 3 m − for all k ≥
1. In particular, Theorem 1.1 is sharp for windmill graphs W m for which (cid:96)/s = 4.Very recently, all graphs which attain the upper estimate in (6.3) were classified in [Ser21].We leave it as an open question whether similar results can be shown for the inequalitiesfrom this paper, especially for Theorem 1.1. References [AlBa19] L. Alon and R. Band,
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