NETWORKS AND HETEROGENEOUS MEDIA doi:10.3934/nhm.2009.4.453 c American Institute of Mathematical Sciences Volume 4, Number 3, September 2009 pp. 453–468 ISOSPECTRAL INFINITE GRAPHS AND NETWORKS AND INFINITE EIGENVALUE MULTIPLICITIES Joachim von Below LMPA Joseph Liouville, FCNRS 2956, Universit´edu Littoral Cˆote d’Opale 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex, France Jose´ A. Lubary Departament de Matem`atica Aplicada II, Universitat Polit`ecnica de Catalunya Jordi Girona, 1–3, 08034 Barcelona, Spain (Communicated by Benedetto Piccoli) Abstract. We consider the continuous Laplacian on infinite locally finite net- works under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a L∞– setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to ℓ∞(Z) is investigated, in particular in trees and periodic graphs. 1. Introduction. It is well known that isospectral finite graphs with respect to the adjacency operator can be non isomorphic, see e.g. [12, 15], as well as isospectral finite networks, i.e. traces of finite topological graphs, with respect to the contin- uous Laplacian can be non isometric and, thereby, the underlying abstract graphs be non isomorphic, see [4, 6]. The node transition conditions for the Laplacian are the continuity at ramification nodes and a Kirchhoff incident flow condition at all nodes. The first aim of the present paper is to analyze the same phenom- ena on infinite uniformly locally finite graphs and networks. The second one is concerned with the existence of eigenvalues of infinite multiplicity, especially the occurrence of inseparable eigenspaces. An essential feature of our approach is the consideration of spaces of bounded functions and of bounded sequences and not a setting in possibly weighted Hilbert Sobolev spaces. It contains the L2–eigenvalue approach, but seems to be more appropriate for spectral links between the network Laplacians and transition or adjacency operators. Often, parts of the continuous and residual spectrum in the L2–setting belong to the point spectrum in spaces of bounded functions. Moreover, the canonical fundamental solutions built by sinus– and cosinus–functions and always belonging to the present setting, can only be treated in Sobolev spaces with sufficiently rapidly decreasing weights. In general, 2000 Mathematics Subject Classification. Primary: 34B45, 05C50; Secondary: 05C10, 35J05, 34L10, 35P10. Key words and phrases. Locally finite graphs and networks, Laplacian, eigenvalue problems, adjacency and transition operators. Jos´eA. Lubary is grateful for partial support by MEC-MTM2005-07660-C02-01, Spain, and for the stay as invited professor to ULCO in Calais in 2006/2007. 453 454 JOACHIM VON BELOW AND JOSE´ A. LUBARY this approach cannot lead to an associated characteristic calculus for the transition operator, while keeping its symmetry properties. The spectrum of the Laplacian on finite networks has been considered by many authors, see e.g. [1, 2, 3, 4, 6, 20, 19, 21, 24] and the references therein. For the infinite case we can refer to [9, 10, 11, 13, 22], for the finite algebraic graph theory to the monographs [12, 14, 15], while for the ℓ2–setting in the infinite case we can refer to [23] and the monograph [27] and the references therein, and for the ℓ∞–setting to [5, 8, 9, 10, 11]. The present paper is organized as follows. Some graph theoretical preliminaries and some results about the eigenvalues of the Laplacian under the aforementioned node transition conditions are summarized in Section 2. Throughout we shall as- sume that all edge lengths are equal to 1. Section 3 is devoted to the notion of isospectrality, to the black hole phenomenon, i.e. the occurrence of inseparable ei- genspaces containing copies of ℓ∞(Z), and to the following result. Theorem 3.2 All infinite uniformly locally finite trees T with finitely many bound- ary vertices, but without ramification nodes of valency 2, are isospectral networks. K More precisely, each λ [0, ) is an eigenvalue of black hole type for ∆T in 2 (T ) L∞(T ). ∈ ∞ − CK ∩ In Section 4 two non isomorphic infinite graphs containing circuits are shown to be isospectral with finite geometric multiplicities, while the associated networks are isospectral and non isometric as well with finite and infinite multiplicities. Section 5 is devoted to the occurrence of black holes in periodic graphs or generalized lattices, in particular to the following results. Corallary 5.1 Any real number λ > 0 satisfying sin √λ = 0 is a black hole ei- genvalue for the Laplacian in each periodic graph of rank m 2 and possesses eigenfunctions of compact support. ≥ Theorem 5.3 Suppose that λ > 0 is an eigenvalue of the Laplacian on a band, i.e. a periodic graph of rank 1. Then λ has infinite geometric multiplicity in 2 ∞ K(T ) L (T ) iff λ is a black hole, or, iff λ possesses eigenfunctions belonging toC 2 (T∩) L∞(T )of finite support. CK ∩ In the final Section 6 examples of families of isospectral non isometric trees, as well as some examples of black hole eigenvalues in periodic graphs are presented. 2. Graphs, networks and Laplacian. For any graph Γ = (V,E, ), the vertex set is denoted by V = V (Γ), the edge set by E = E(Γ) and the incidence∈ relation by V E. The valency of each vertex v is denoted by γ(v) = card e E v e . Unless∈⊂ × otherwise stated, all graphs considered in this paper are assumed{ ∈ to∈ be} nonempty, simple, connected and uniformly locally finite, i.e. max γ(v) =: γmax < . (1) v∈V (Γ) ∞ The simplicity property means that Γ contains no loops, and at most one edge can join two vertices in Γ. Moreover, the conditions imply that Γ is countable. For a given numbering of the vertices vi, i N, set γi = γ(vi) and define the adjacency ∈ ISOSPECTRAL INFINITE GRAPHS AND NETWORKS 455 matrix or adjacency operator by V (Γ) V (Γ) (Γ) = (aih) : R R , (2) A i,h∈N −→ where 1 if vi and vh are adjacent in Γ, aih = (0 else. Note that (Γ) is indecomposable iff Γ is connected. Moreover, with the notation ℓp(Γ) := ℓpA(V (Γ)), (Γ) maps ℓp(Γ) into ℓp(Γ) for each p [1, ]. The same holds for the row–stochasticA transition matrix or transition operator∈ ∞defined by = Diag ( (Γ) e)−1 (Γ) = Diag γ−1 (Γ), (3) Z A A i i A where e denotes the sequence with entries equal to 1. For a subgraph Θ in Γ let Θ¯ = (V (Θ), E(Θ)¯ , ) denote the subgraph of Γ spanned by the vertices in Θ with ∈ E(Θ)¯ = e e E(Γ), e V (Γ) V (Θ) . { ∈ ∩ ⊂ } The subgraph Θ is called induced if Θ¯ = Θ. Two subgraphs are called disjoint if they have no vertex in common, and essentially disjoint if they have only a finite number of edges in common. The (combinatorial) distance between two vertices v1 and v2 is defined as the minimal number of edges of all paths joining v1 and v2. For further graph theoretical terminology we refer to [16, 25, 26]. Moreover, without loss of generality, we consider each graph as a connected topological graph in Rm, i.e. V (Γ) Rm, and the edge set consists in a collection of ⊂ m Jordan curves E(Γ) = πj : [0, 1] R j N all of length 1 with the following { → ∈ } properties: Each support ej := πj ([0, 1]) has its endpoints in the set V (Γ), any two vertices in V (Γ) can be connected by a path with arcs in E(Γ), and any two edges ej = eh satisfy ej eh V (Γ) and card(ej eh) 1. The arc length parameter 6 ∩ ⊂ ∩ ≤ of an edge ej is denoted by tj . The trace of the graph Γ = (V,E, ) defines its associated network ∈ G = πj ([0,lj]) , j∈N [ ν ν m that is called a -network, if all πj ([0, 1], R ). Throughout we shall assume that at least C ∈C 2 m j N : πj ([0, 1], R ). ∀ ∈ ∈C Thus, endowed with the induced topology of Rm, G is a connected and locally m compact space in R . We shall distinguish the boundary vertices Vb = vi { ∈ V γi =1 from the ramification nodes Vr = vi V γi 2 , especially, we define } { ∈ ≥ } the essential ramification nodes by V = vi V γi 3 . ess { ∈ ≥ } Two networks G1 and G2 are isometric (G1 ∼= G2) if there is an homeomorphism H : G1 G2 such that for each edge e G1, H e is an isometric diffeomorphism onto some→ edge of G . In particular, H is⊂ length preserving. Moreover, the underly- 2 ing abstract graphs Γ1 and Γ2 are called isomorphic as graphs (Γ1 Γ2) if there is a bijection V (Γ ) V (Γ ) that preserves the adjacency relation between≃ vertices. 1 −→ 2 If G1 and G2 are isometric networks, then the underlying abstract graphs Γ1 and Γ2 are isomorphic as graphs. In fact, in the present case of equal edge lengths, this is an equivalence: G = G Γ Γ (4) 1 ∼ 2 ⇐⇒ 1 ≃ 2 The orientation of the graph Γ is given by the incidence matrix or incidence operator E(Γ) V (Γ) (Γ) = (dik) : R R (5) D i,k∈N −→ 456 JOACHIM VON BELOW AND JOSE´ A. LUBARY with 1 if πj (1) = vi, dij = 1 if πj (0) = vi, − 0 otherwise. By (1), (Γ) is a bounded operator ℓ∞(E(Γ)) ℓ∞(V (Γ)).