OOn extremal eigenvalues of the graph Laplacian
Andrea Serio ∗ Abstract
Upper and lower estimates of eigenvalues of the Laplacian on a metric graph havebeen established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo.Both these estimates can be achieved at the same time only by highly degenerateeigenvalues which we call maximally degenerate . By comparison with the maximaleigenvalue multiplicity proved by I. Kac and V. Pivovarchik in 2011 we characterizethe family of graphs exhibiting maximally degenerate eigenvalues which we call lassotrees , namely graphs constructed from trees by attaching lasso graphs to some of thevertices.
Overview
We are interested in the study of two bounds of the eigenvalues of the graph Laplacianproven in [1] which depend on a few simple geometrical and topological properties of thegraph, namely the total length L , the number of Dirichlet and Neumann pendant vertices D and N , and the first Betti number β . λ n ≥ m n = (cid:40) π L n if n < N + β, π L (cid:0) n − N + β (cid:1) if n ≥ N + β n ≥ λ n ≤ M n = π L (cid:18) n − D + N + β β (cid:19) n ∈ N , (2)If D (cid:54) = 0 then the lower bound estimate holds for all n ∈ N . Otherwise if D = 0 then λ = 0.In [2] it was shown that (2) is attained by an infinite sequence of eigenvalues { µ n i = M n i } i ∈ N generated by a family of graphs with D = N = 0 and any β ≥
2, or β = D + N =1, called respectively Windmill graphs, Neumann lasso graph and Dirichlet lasso graph.It can be observed that these graphs also provide examples of sequences of eigenvalueswhich exhibit the equality in (1), { ν n j = m n j } j ∈ N . Remarkably, the sequences µ n i and ν n j coincide. This is possible because the indices n i and n j are respectively the smallest andthe largest of a sequence of degenerate eigenvalues with multiplicity m : i = j ⇐⇒ µ n j = ν n i ⇐⇒ n j − n i = m − . (3) ∗ Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden.Email: [email protected] . a r X i v : . [ m a t h . SP ] A ug e call lower sharp and upper sharp eigenvalues those which satisfy the equality in(1) and (2) respectively, and in general we call (degenerate) sharp eigenvalues theeigenvalues of multiplicity m ≥ m ≥
2) with smallest index upper sharp and largestindex lower sharp like those in equation (3). The purpose of this work is to investigatewhich graphs exhibit sharp eigenvalues and discuss their properties. The text is organizedinto three sections: Introduction and notation, Properties of sharp eigenvalues, and Mainresults. We can summarize our findings as follows.In Proposition 2.2, by direct comparison of (1) and (2), we obtain an upper boundfor the maximal eigenvalue multiplicity m U = m U ( G ) = D + N + 2 β −
1. Eigenvalues ofmultiplicity m U are called maximally degenerate eigenvalues .In Theorem 2.5 we show that sharp eigenvalues are characterized by being maximallydegenerate. This is used to show that sharp eigenvalues are preserved when multiplegraphs are joined together at one of their Dirichlet pendant vertices (Lemma 3.1) or whena loop graph—with certain prescribed length—is attached to any Neumann pendant vertex(Lemma 3.3).In the proof of the main result, Theorem 3.4, we show how the aforementioned Lem-mata can be used to construct a graph with arbitrary N , D , β which produce sequences ofdegenerate sharp eigenvalues. Graphs which can be constructed by recursive applicationsof Lemmata 3.1 and 3.3 are trees where some of the pendant vertices have a loop graphattached, or, equivalently, trees decorated with some lasso graphs (also called tadpole orlollipop graphs), for this reason we call them lasso trees .By comparing m U with the maximal eigenvalue multiplicity m M proved in [3] we observethat m U ( G ) ≥ m M ( G ) with the equality occurring if and only if G is a lasso tree. Finally weconclude in Theorem 3.6 that sharp eigenvalues appear only in the spectra of lasso trees. In this section we present the essentials about metric graphs used in the present text, fora more general introduction we refer to [4] and [5], see also [6] and the recent [7].
Metric graphs.
Metric graphs are constructed as the quotient space of a set of dis-tinct real intervals under an equivalence relation on the set of their endpoints. Let E = (cid:116) i e i , e i := [ x i − , x i ] be the disjoint union of closed real intervals and let ∼ bean equivalence relation over the endpoints of E . The quotient space G = E/ ∼ is a metricgraph, whose set of edges and vertices are E = E ( G ) and, respectively, V = V ( G ) = (cid:116) i { x i − , x i } / ∼ . The metric and measure over G are inherited from the Euclidean metricand Lebesgue measure over the edges. Moreover we denote by L = (cid:80) i | x i − x i − | thetotal length of G . Since we are interested in metric graphs which are compact and withfinite total length, we assume that G has a finite number of edges, | E | < ∞ , each of themcompact. We allow the presence of loops and multiple edges. Cycles and the first Betti number.
A cycle is a finite sequence of distinct edges { e ( j ) } nj =1 associated to a sequence of distinct vertices { v ( j ) } nj =1 such that • e ( j ) is incident to v ( j ) and v ( j +1) for j = 1 , . . . , n − • e ( n ) is incident to v ( n ) and v (1) . 2 cycle of length 1 is also called loop. A graph with just one edge which is also a loop iscalled loop graph.We denote by β = β = | E | − | V | + 1, the first Betti number of G ; β coincides with thecircuit rank of G , namely the least number of edges that need to be removed in order toturn G into a tree. Functions on metric graphs.
Let L ( G ) := (cid:76) i L [ x i − , x i ]. The space L ( G ) equippedwith the inner product (cid:104) f, g (cid:105) := (cid:80) i (cid:82) e i f g dx is a well defined Hilbert space. If f ∈ L ( G )is continuously differentiable over the edge e i = [ x i − , x i ], the oriented derivatives of f at the endpoints of the interval e i are defined by ∂f ( x i − ) = f (cid:48) ( x i − ) and ∂f ( x i ) = − f (cid:48) ( x i ). The oriented derivatives are well defined for functions in the Sobolev space H ( G ) := (cid:76) i H [ x i − , x i ]. Laplacian, vertex conditions, and eigenvalues.
Given a subset of the pendant ver-tices D , which we call Dirichlet vertices, we define the Laplacian operator L = L ( G , D ) := − d dx with domain D ( L ) = D ( G , D ) as the set of functions f ∈ H ( G ) subject to thefollowing conditions: • f is continuous at the vertices, so f ∈ C ( G ) (continuity condition), • the oriented derivatives of f sum to zero at each non Dirichlet vertex: (cid:80) x j ∈ v ∂f ( x j ) = 0 ∀ v ∈ V \ D (Kirchhoff condition), • f vanishes at the Dirichlet vertices, f ( v ) = 0 ∀ v ∈ D (Dirichlet condition).The continuity and Kirchhoff conditions together are called standard vertex conditions(in the literature sometimes called natural).Standard vertex conditions at pendant vertices v / ∈ D read as Neumann conditions: f (cid:48) ( v ) = 0; hence we call these vertices Neumann and denote their set by N . We denote bythe corresponding calligraphic letter the cardinality of the two different type of pendantvertices N = | N | , D = | D | .Standard vertex conditions at a vertex v of degree two read as continuity of both thefunction and its derivative. Let G have a pair of edges e i , e j incident to a vertex v of degreetwo and let G (cid:48) be the graph where e i , e j are replaced by a single edge with length equal tothe sum of the lengths of e i , e j . Then G (cid:48) , L ( G (cid:48) ), and D ( G (cid:48) , D ) are, respectively, isomorphicto G , L ( G ), and D ( G , D ). Hence, degree two vertices play no role in the study and can befreely removed whenever they occur in the construction of graphs.Under the above hypothesis L is a self-adjoint unbounded positive operator whosespectrum is exclusively discrete with unique accumulation point at + ∞ , [4]. We denotethe spectrum by σ ( L ) = { λ n } n ∈ N . Whenever we say that an indexed eigenvalue λ n hasmultiplicity m we assume n to be the smallest index of the degenerate eigenvalue, i.e. λ n = · · · = λ n + m − . Moreover, unless differently stated, we shall assume G to be connected.Under this hypothesis follows that the ground state λ is simple (see [8]) and λ = 0 if andonly if D = ∅ . In [1] the authors show that if G is not a loop graph then the eigenvaluesof L ( G , D ) satisfy the inequalities (1), (2). Quantum Graphs.
The term quantum graph is used in general to refer to the tripleΓ = ( G , − d dx + q ( x ) , D ( G , D )). In the present setup q ≡
0, so a metric graph G togetherwith a set of Dirichlet vertices D suffice to determine a quantum graph Γ. Hence we usethe generic term graph to refer to both metric and quantum graph whenever the set of3irichlet vertices and the associated Laplacian are clear from the context. In particularwe may speak of the eigenvalues of a graph meaning the eigenvalues of the associatedLaplacian.In the study of the spectral estimates of quantum graphs several techniques have beendeveloped, many of which put in relation modifications of the metric graph with the changesoccurring in the spectrum. Several of these are discussed in [9] under the name of surgeryprinciples . Lemmata 3.1 and 3.3 make use of a particular case of two such principles. Inline with the previous paragraph, any modification brought to a graph G and its set ofDirichlet vertices D should be reflected in the associated Laplacian. We start by making a general observation.
Observation . The sequences { m n } n ∈ N and { M n } n ∈ N given by (1) and (2) are strictlyincreasing. As a consequence of this, if λ n = m n then λ n < m n +1 ≤ λ n +1 ; hence • if λ n is lower sharp then λ n < λ n +1 .Similarly λ n = M n implies that • if λ n is upper sharp then λ n − < λ n . Proposition 2.2.
The multiplicity of any eigenvalue of the graph Laplacian is at most D + N + 2 β − . Eigenvalues with maximal multiplicity are called maximally degenerateeigenvalues.Proof. Consider λ n be a degenerate eigenvalue of multiplicity m ≥
2, so n ≥ λ n and λ n + m − we have m n + m − ≤ M n . (4)Assuming n + m − ≥ N + β , then (4) implies m ≤ D + N + 2 β − n + m − < N + β then (4) implies m ≤ n + 2 D + N + 3 β − m < D + N + 2 β − Lemma 2.3.
Let λ n be a degenerate eigenvalue of multiplicity m ≥ . Then λ n is a sharpdegenerate eigenvalue if and only if λ n is maximally degenerate m = D + N + 2 β − .Proof. Because of the simplicity of the ground state, n ≥
2. Assume n + m − ≥ N + β ;hence by definition m n + m − = ( π/ L ) ( n + m − − ( N + β ) / and M n = m n + m − ⇐⇒ n − D + N + β β = n + m − − N + β ⇐⇒ m = D + N + 2 β − . Assume instead n + m − < N + β ; hence m n + m − = ( π/ L ) ( n + m − /
4; we show thatthis is not compatible with the hypothesis. In fact M n = m n + m − ⇐⇒ n − D + N + β β = n + m −
12 (6) ⇐⇒ m = n + 2 D + N + 3 β − , therefore n + m − < N + β reads n + D + β < n = 1, a contradiction.4he proof of Lemma 2.3 suggests there might exist simple eigenvalues which are bothupper sharp and lower sharp at the same time, which shall be called simple sharp eigen-values.Consider first n ≥
2: if M n = m n then either(i) n < N + β , so by (6) n = 1, which is excluded,(ii) n ≥ N + β , so by (5) D + N + 2 β = 2.If we consider n = 1 then either(iii) D = 0 thus λ = 0 = M , so N + 3 β = 2,(iv) D (cid:54) = 0 and 1 < N + β , so by (6) 2 D + N + 3 β = 3,(v) D (cid:54) = 0 and 1 ≥ N + β , so by (5) D + N + 2 β = 2.Case (ii) is satisfied by any of the following: • β = 1 and N = D = 0, i.e. the loop graph which should be disregarded as it is anexceptional case for which neither (1) nor (2) holds. • β = 0 and N + D = 2, i.e. the interval with any admissible vertex conditions at itsendpoints.Case (iii) implies N = 2 , β = 0, namely the Neumann-Neumann interval. Case (iv) doesnot have solutions. Case (v) implies β = 0 and either N = 0 , D = 2, or N = D = 1 hencethe remaining two possible vertex conditions for the single interval.Therefore the single interval with any of the admissible vertex conditions provides theonly three examples of graphs with simple sharp eigenvalues, and in particular the wholespectrum is composed only by simple sharp eigenvalues: • N = 2 , D = 0, the spectrum is λ NNn = π L ( n − , • N = 1 , D = 1, the spectrum is λ NDn = π L ( n − ) , • N = 0 , D = 2, the spectrum is λ DDn = π L n . Proposition 2.4.
An eigenvalue is simple sharp if and only if the underlying graph is asingle interval with any of the three possible combinations of vertex conditions listed above.
From now on we refer to as sharp eigenvalues the eigenvalues which are either simpleor degenerate eigenvalues which are sharp. The above discussion can be summarized bythe following statement:
Theorem 2.5.
Let G be a graph which is not a cycle and let λ be an eigenvalue of someLaplacian over G . Then λ is sharp if and only if it is maximally degenerate. .2 Sharp eigenvalues and fully supported eigenspace In this section we show a necessary property of the eigenfunctions associated to degeneratesharp eigenvalues. For its proof we need the following proposition about the regularity ofthe eigenvalues seen as functions dependent on the length of an edge of the graph.
Proposition 2.6.
For any fixed index n ∈ N and edge e of length (cid:96) , the function (cid:96) (cid:55)→ λ n ( (cid:96) ) is continuous on (0 , + ∞ ) .Proof. Consider the Courant-Fischer eigenvalues characterizations via the Rayleigh quo-tient λ n = min X ⊂ D q dim( X )= n max u ∈ X (cid:107) u (cid:48) (cid:107) (cid:107) u (cid:107) = (cid:107) ψ (cid:48) n (cid:107) , (7)where D q = D q ( G , D ) := { f ∈ H ( G ) ∩ C ( G ) : f ( v ) = 0 ∀ v ∈ D } and ψ n is any normalizedeigenfunction associated to λ n . In order to prove the statement we show that ρ (cid:55)→ λ n ( ρ(cid:96) )is continuous in ρ = 1. Let G ρ be the modification of G where the edge e is stretched by afactor ρ , i.e. e is replaced by ρe and consequently (cid:96) replaced by ρ(cid:96) . Let X ⊂ D q be anysubset realizing the minimum in (7). Let X ρ ⊂ D q ( G ρ ) be the space obtained from X bystretching each function over the edge e , i.e. f ρ ( x ) = f ( x ) if x ∈ G ρ \ ρe and f ρ ( x ) = f ( x/ρ )if x ∈ ρe . From the Rayleigh quotient it follows that λ n ( ρ(cid:96) ) ≤ max u ρ ∈ X ρ (cid:107) u (cid:48) ρ (cid:107) (cid:107) u ρ (cid:107) . (8)We compute (cid:107) u (cid:48) ρ (cid:107) L ( G ρ ) = (cid:90) G ρ \ ρe ( u (cid:48) ρ ) dx + (cid:90) ρe ( u (cid:48) ρ ) dx = (cid:90) G\ e ( u (cid:48) ) dx + (cid:90) e (cid:18) ρ u (cid:48) (cid:19) ρ dx = (cid:90) G ( u (cid:48) ) dx + (cid:18) ρ − (cid:19) (cid:90) e ( u (cid:48) ) dx (9)and similarly we also obtain (cid:107) u ρ (cid:107) L ( G ρ ) = (cid:90) G u dx + ( ρ − (cid:90) e u dx. (10)Therefore, if ρ ≤ u ρ ∈ X ρ (cid:107) u ρ (cid:48) (cid:107) (cid:107) u ρ (cid:107) = max u ∈ X (cid:107) u (cid:48) (cid:107) L ( G ) + ( ρ − (cid:107) u (cid:48) (cid:107) L ( e ) (cid:107) u (cid:107) L ( G ) + ( ρ − (cid:107) u (cid:107) L ( e ) ≤ ρ max u ∈ X (cid:107) u (cid:48) (cid:107) L ( G ) (cid:107) u (cid:107) L ( G ) ≤ ρ λ n ( (cid:96) ) . (11)Moreover, from the monotonicity of the eigenvalues (see for example Corollary 3.12 in [9])we know that ρ ≤ ⇒ λ n ( (cid:96) ) ≤ λ n ( ρ(cid:96) ).Thus for ρ ≤ λ n ( (cid:96) ) ≤ λ n ( ρ(cid:96) ) ≤ ρ λ n ( (cid:96) ) . (12)6y changing (cid:96) with (cid:96)/ρ then we can deduce the more general inequality for any ρ > (cid:8) , ρ − (cid:9) λ n ( (cid:96) ) ≤ λ n ( ρ(cid:96) ) ≤ max (cid:8) , ρ − (cid:9) λ n ( (cid:96) ) , (13)which shows the continuity in ρ = 1 of ρ (cid:55)→ λ n ( ρ(cid:96) ) and hence the claimed continuity of (cid:96) (cid:55)→ λ n ( (cid:96) ) for (cid:96) ∈ (0 , + ∞ ).The next lemma shows that both upper and lower sharp eigenvalues can be associatedto eigenfunctions that do not identically vanish on any edge of the graph. Lemma 2.7. If λ n is either lower or upper sharp then for each edge e there exists aneigenfunction associated to λ n which is not identically zero on e .Proof. Assume λ n = M n and let (cid:96) ( e ) = (cid:96) . Making e longer increases the total length of thegraph and consequently decreases M n , we write M n ( (cid:96) ) to highlight the estimate dependenceon the length of the edge e . In order not to violate (2), λ n must also decrease, at leastas much as its upper bound. Assume λ n has multiplicity m ; hence by Observation 2.1 λ n − < λ n = · · · = λ n + m − < λ n + m . By Proposition 2.6 all eigenvalues are continuousfunctions in the length (cid:96) ( e ), so there exists ε > ∀ (cid:96) ∈ [ (cid:96) , (cid:96) + ε ] λ n − ( (cid:96) ) < λ n ( (cid:96) ) and λ n + m − ( (cid:96) ) < λ n + m ( (cid:96) ) . (14)Let { ψ j } n + m − j = n be a basis of the m -dimensional eigenspace associated to λ n . If each ψ j isidentically zero on e , then ψ j is still an eigenfunction after perturbing the length of e overthe interval [ (cid:96) , (cid:96) + ε ] and by the Rayleigh quotient it is associated to an eigenvalue equalto λ n ( (cid:96) ) with the same multiplicity m . Because of (14) the indices of the eigenvalues arepreserved; hence λ j ( (cid:96) ) ≡ λ j ( (cid:96) ) ∀ (cid:96) ∈ [ (cid:96) , (cid:96) + ε ] . (15)This leads to the following contradiction M n ( (cid:96) + ε ) < M n ( (cid:96) ) = λ n ( (cid:96) )= λ n ( (cid:96) + ε ) ≤ M n ( (cid:96) + ε ) . (16)Thus there exists an eigenfunction not identically zero on e .We then have the next corollary. Corollary 2.8. If λ is a sharp eigenvalue then there exists an eigenfunction associated to λ which does not identically vanish on any edge of the graph. The main theorem is proven in a constructive manner and relies upon the next two lemmata,each of them providing an operation which preserves sharp eigenvalues. The first of them,Lemma 3.1, tells us that joining together graphs which share a sharp eigenvalue preservesnot only the eigenvalue but also its sharpness.Let {G i } pi =1 be a finite set of graphs, each of them with N i Neumann, D i (cid:54) = 0 Dirichletpendant vertices and β i first Betti numbers respectively. For each G i fix a Dirichlet vertex v i ∈ D i (see for example the set of graphs on the left of figure 1). Assume that thespectrum of each G i contains the same eigenvalue λ , not necessarily with the same index λ n i ( G i ) = λ ∀ i . Consider the graph G obtained by the disjoint union of all graphs (cid:70) pi =1 G i with the vertices v i replaced by a single vertex endowed with standard vertex conditionsas in Figure 1. Then λ is still an eigenvalue of G . We have then the following statement.7 emma 3.1. If λ n i = λ is a sharp eigenvalue of each G i , then λ n = λ, n = 2 − p + (cid:80) pi =1 n i is also a sharp eigenvalue of G . Figure 1: Example of application of Lemma 3.1 to three graphs (left) joined at one chosenDirichlet vertex for each of them in order to obtain the graph on the right. Here and in thefollowing figures the symbol ◦ stands for a Dirichlet pendant vertex and • for a Neumannpendant vertex.The above lemma allows us to build trees with sharp eigenvalues: one starts by joiningintervals into star graphs with at least one Dirichlet pendant and then the star graphs intoa tree. This Lemma allows the construction of three graphs with any prescribed numberof Dirichlet and Neumann pendant vertices which exhibit sharp eigenvalues. It remainsto show that it is as well possible to prescribe the first Betti number and still be able toconstruct a graph with sharp eigenvalues. This is achieved by Lemma 3.3 which showsthat sharp eigenvalues are preserved after attaching a cycle to a Neumann pendant. Wehave already mentioned that the loop graph is the only graph which does not satisfy theinequalities (1 2), in particular we can notice the following: Proposition 3.2.
The spectrum of the loop graph L L of length L is given by σ ( L L ) = { λ = 0 } ∪ (cid:110) λ j = λ j +1 = π L (2 j ) : j ∈ N > (cid:111) . The even eigenvalues of the loop graphexceeds the upper estimate (2) by a term + as follows: λ j ( L L ) = π L (cid:18) j − (cid:19) . (17) The odd eigenvalues, excluded the first, differs from the lower estimate (1) by a term − as follows: λ j +1 ( L L ) = π L (cid:18) j + 1 − − (cid:19) . (18)Therefore, given λ > j ∈ N , the loop graph with length (cid:96) := 2 jπ/ √ λ hasthe eigenvalue λ j = λ with multiplicity 2. Now consider G any graph with at least oneNeumann pendant vertex v with a certain eigenvalue λ n ( G ) = λ . Let G v be the graphobtained by attaching the loop graph of length (cid:96) to the Neumann vertex v with standardvertex conditions imposed there as in Figure 2. We have the following statement: Lemma 3.3. If λ n = λ is a sharp eigenvalue of G then λ n v = λ, n v = n + 2 j − is asharp degenerate eigenvalue of G v . In particular, the multiplicity of λ going from G to G v increases by one. G and the loop graph, on the right the graph G v .Lemmata 3.1 and 3.3 applied to a set of intervals and loop graphs provide the toolsto derive the main result, Figure 3 shows an example of graph constructed following theproof of Theorem 3.4. Theorem 3.4.
Given N , D , β ∈ N ∪ { } such that N + D + β ≥ , there exists a graphwith N Neumann, D Dirichlet pendant vertices respectively and first Betti number β whichexhibits an infinite sequence of sharp eigenvalues. Figure 3: Example of graph constructed via Theorem 3.4 with D = 4 , N = 2 , β = 2 withthe lengths of the edges to scale.In [3] the authors show that the maximal multiplicity of eigenvalues of the Schr¨odingeroperator − d dx + q ( x ) with potential q ∈ L defined on a compact graph G is m M = β + P T −
1, where P T is the number of pendant vertices of the tree graph T G obtainedfrom G after contracting each cycle to a vertex.We observe that the contraction of any cycle may generate at most one new pendantvertex, thus P T − ( D + N ) ≤ β . This means that m M ≤ m U with the equality occurringif and only if T G has exactly β pendant vertices more than G , or equivalently G is a lassotree. Definition 3.5.
A lasso tree is a compact metric graph where each cycle is a loop incidentto a vertex of degree three.The previous observation together with Theorem 2.5 lead us to the following conclusion.
Theorem 3.6. If G is a metric graph with sharp eigenvalues, then G is a lasso tree.Remark . We point out that Lemmata 3.1 and 3.3 can be applied recursively to constructlasso trees, with any possible topological structure, having sharp eigenvalues.9igure 4: Example of a lasso tree.
Proof of Lemma 3.1.
By Theorem 2.5 each λ n i is maximally degenerate; hence with multi-plicity m i = D i + N i + 2 β i −
1. The spectrum of the disjoint union of the graphs {G i } is thedisjoint union of their eigenvalues, therefore λ is an eigenvalue of (cid:70) pi =1 G i with multiplicity (cid:80) m i . Since λ n i − < λ n i then there are ( (cid:80) n i ) − p strictly smaller eigenvalues than λ ,possibly zero. Hence the smallest index of λ on (cid:70) pi =1 G i is 1 + (cid:80) ( n i − { v i } by a single vertex v endowed with standard vertex conditions is an operationwhich increases the dimension of the domain of the quadratic form associated to the Lapla-cian by one, thus it corresponds to a rank one perturbation of the operator which pushesall the eigenvalues down, but no further than one index, i.e. it interlaces the eigenvalues λ j − ( (cid:70) pi =1 G i ) ≤ λ j ( G ) ≤ λ j ( (cid:70) pi =1 G i ) ∀ j ∈ N . This operation can be seen as the inverseof a particular case of Theorem 3.4 (2) in [9], see also Theorem 3.1.8 in [4]. Since λ hasmultiplicity (cid:80) m i on (cid:70) pi =1 G i after the change of vertex condition, λ is still an eigenvalueon G , with multiplicity at least m = ( (cid:80) m i ) − n = 2 + (cid:80) ( n i − n and m are indeed exact. Notice thatwhen going from (cid:70) pi =1 G i to G we have that • the number of Dirichlet pendant vertices is reduced by p , D = ( (cid:80) pi =1 D i ) − p ; • the number of Neumann pendant vertices is preserved, N = (cid:80) pi =1 N i ; • the first Betti number is preserved, β = (cid:80) pi =1 β i ;Therefore we compute that m = (cid:16)(cid:88) m i (cid:17) − n (cid:88) i =1 ( D i + N i + 2 β i − − D + N + 2 β − λ must have preciselymultiplicity m and consequently lowest index n . By Theorem 2.5 λ n must be a sharpeigenvalue. 10 roof of Lemma 3.3. Consider the spectrum of the disjoint union of G and L (cid:96) , which is thedisjoint union of their spectra. Then λ has now smallest index n v = ( n −
1) + (2 j −
1) + 1and if λ has multiplicity m on G then its multiplicity on G (cid:116) L (cid:96) is m + 2. The action ofattaching the loop graph to G at the vertex v ∈ G is a rank one perturbation of the graphLaplacian which decreases the domain of its associated quadratic form and consequentlypushes the eigenvalues up, but no further than the eigenvalue of next index; hence λ = λ n +2 j − ( G (cid:116) L (cid:96) ) ≤ λ n +2 j − ( G v ) ≤ λ n +2 j ( G (cid:116) L (cid:96) ) = λ. (20)We now show that after this operation the multiplicity of λ is reduced by one, i.e. it is m + 1, and consequently the smallest index of λ on G v is still n v . Let us parameterize theloop graph by the interval [ − (cid:96)/ , (cid:96)/
2] with the zero placed in v . Any eigenfunction ϕ on G can be extended to G v by (cid:101) ϕ ( x ) := (cid:40) ϕ ( x ) if x ∈ G ,ϕ ( v ) cos( √ λx ) if x ∈ [ − (cid:96)/ , + (cid:96)/ . (21)So all the eigenfunctions on G associated to λ n are embedded in G v . In addition thefollowing eigenfunction (cid:101) ϕ from L (cid:96) can be embedded in G v (cid:101) ϕ ( x ) := (cid:40) x ∈ G , sin( √ λx ) if x ∈ [ − (cid:96)/ , + (cid:96)/ . (22)Hence going from G (cid:116) L (cid:96) to G v the multiplicity of λ is reduced by one. Now notice that byTheorem 2.5 the multiplicity of λ on G is m = D + N + 2 β −
1, and hence m + 1 = D + ( N −
1) + 2( β + 1) − . (23)which is the maximal admissible eigenvalue multiplicity on G v since this graph has onemore cycle and one less Neumann pendant than G . Again by Theorem 2.5, λ must be asharp degenerate eigenvalue of G v with smallest index necessarily n v . Proof of Theorem 3.4.
Let N , D and β be given. In order to construct a graph with thesecorresponding numbers of Neumann pendants, Dirichlet pendants and first Betti numberrespectively, it is enough to consider • N + β copies of Neumann-Dirichlet intervals I ND of length (cid:96) N = π/ • D copies of Dirichlet-Dirichlet intervals I DD of length (cid:96) D = π , • β copies of loop graph L of length (cid:96) L = 2 π .Notice that the above three graphs share the following sequence of eigenvalues: λ NDj = λ DD j − = λ L j − = (2 j − . (24)Apply Lemma 3.1 to all the above intervals, both Neumann-Dirichlet and Dirichlet-Dirichletto deduce that { λ n j = (2 j + 1) } j ∈ N is a sequence of sharp eigenvalues, each of multiplicity N + β + D −
1, where n j = 2 − (( N + β ) + D ) + ( N + β ) · j + D · (2 j − − ( N + β ) + ( N + β + 2 D ) j. (25)11otice that the length of the loop graph L can be rewritten as (cid:96) L = 2 π = π · j − (cid:113) λ L j − . (26)Therefore we can recursively apply Lemma 3.3 β number of times and obtain the newsequence of sharp eigenvalues { λ (cid:101) n j = (2 j − } , each of multiplicity N + D + 2 β −
1, whichis maximal, where (cid:101) n j = n j + 2(2 j − β − β = 2 − ( N + 4 β ) + ( N + 4 β + 2 D ) j. (27) Observation . The proof of Theorem 3.4 with N = D = 0 recovers the family ofWindmill graphs defined [2]. Acknowledgement
The author wishes to thank Pavel Kurasov for valuable suggestions and guiding, and JacobMuller.
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