Spectral Estimates for Infinite Quantum Graphs

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.


Introduction
The main focus of our paper is on the study of spectra of quantum graphs. The notion of "quantum graph" refers to a graph G considered as a one-dimensional simplicial complex and equipped with a differential operator. The spectral and scattering properties of Schrödinger operators on such structures attracted a considerable interest during the last two decades, as they provide, in particular, relevant models of nanostructured systems (we only mention recent collected works and monographs with a comprehensive bibliography: [9], [10], [23], [51]).
Let G be a locally finite connected metric graph, that is, a locally finite connected combinatorial graph G d = (V, E), where each edge e ∈ E is identified with the interval (0, |e|) and | · | denotes the edge length. We shall always assume throughout the paper that each edge has finite length, that is, | · | : E → (0, ∞). In the Hilbert space L 2 (G) = e∈E L 2 (e), we can define the Hamiltonian H which acts in this space as the (negative) second derivative − d 2 dx 2 e on every edge e ∈ E. To give H the meaning of a quantum mechanical energy operator, it must be self-adjoint and hence one needs to impose appropriate boundary conditions at the vertices. Kirchhoff (also known as Kirchhoff-Neumann) conditions (2.6) are the most standard ones (cf. [10]) and the corresponding operator denoted by H is usually called a Kirchhoff (Kirchhoff-Neumann) Laplacian (we refer to Sections 2.2-2.4 for a precise definition of the operator H). If the graph G is finite (G has finitely many vertices and edges), then the spectrum of H is purely discrete (see, e.g., [10]). During the last few years, a lot of effort has been put in estimating the first nonzero eigenvalue of the operator H (notice that 0 is always a simple eigenvalue if G d is connected) and also in understanding its dependence on various characteristics of the corresponding metric graph including the number of essential vertices of the graph (vertices of degree 2 are called inessential); the number or the total length of the graph's edges; the edge connectivity of the underlying (combinatorial) graph, etc. For further information we refer to a brief selection of recent articles [3], [4], [8], [38], [39], [42], [52].
If the graph G is infinite (there are infinitely many vertices and edges), then the corresponding pre-minimal operator H 0 defined by (2.7) is not automatically essentially self-adjoint. One of the standard conditions to ensure the essential selfadjointness of H 0 is the existence of a positive lower bound on the edges lengths, ℓ * (G) = inf e∈E |e| > 0 (see [10]). Only recently several self-adjointness conditions without this rather restrictive assumption have been established in [24], [41] (see Section 2.3 for further details). Of course, the next natural question is the structure of the spectrum of the operator H. Clearly, the spectrum of an infinite quantum graph is not necessarily discrete and hence one is interested in the location of the bottom of the spectrum, λ 0 (H), as well as of the bottom of the essential spectrum, λ ess 0 (H), of H. Since the graph is infinite, many quantities of interest for finite quantum graphs (e.g., the number of vertices, edges, or its total length) are no longer suitable for these purposes and the corresponding bounds usually lead to trivial estimates. However, it is widely known that quantum graphs in a certain sense interpolate between Laplacians on Riemannian manifolds and difference Laplacians on combinatorial graphs and hence quantum graphs can be investigated by modifying techniques that have been developed for operators on manifolds and graphs and we explore these analogies in the present paper. Notice that this insight has already proved to be very fruitful and it has led to many important results in spectral theory of operators on metric graphs (see, e.g., [10]). Although quantum graphs are essentially operators on one dimensional manifolds, our point of view is that the corresponding results and estimates should be of combinatorial nature.
Our central result is a Cheeger-type estimate for quantum graphs, which establishes lower bounds for λ 0 (H) and λ ess 0 (H) in terms of the isoperimetric constant α(G) of the metric graph G (Theorem 3.4). Although the Cheeger-type bound for (finite) quantum graphs was proved 30 years ago by S. Nicaise (see [46,Theorem 3.2]), we give a new purely combinatorial definition of the isoperimetric constant (see Definition 3.2) and hence, on the one hand, this improves the corresponding result from [46] (cf. Lemma 3.7) and, on the other hand, establishes a connection with isoperimetric constants for combinatorial graphs (see Lemma 4.1 and also (4.10)-(4.11)). To a certain extent this connection is expected (cf. Theorem 2.12 and also [6,14,11,51]). Moreover, it was observed recently in [24,41] by using the ideas from [40] that spectral properties of the operator H are closely connected with the corresponding properties of the discrete Laplacian defined in ℓ 2 (V; m) by the expression where the weight function m : V → R >0 is given by Using this connection, several criteria for λ 0 (H) and λ ess 0 (H) to be positive have been established in [24], however, in terms of isoperimetric constants and volume growth of the combinatorial graphs introduced, respectively, in [5] and [26], [32] (in this paper we obtain these results as simple corollaries of our estimate (3.8)).
Despite the combinatorial nature of (3.3) and (3.4), it is known that computation of the combinatorial isoperimetric constant is an NP-hard problem (see [33,35] for further details). Motivated by [5] and [20], we introduce a quantity, which sometimes is interpreted as a curvature of a graph, leading to estimates for the isoperimetric constants α(G) and α ess (G). It also turns out to be very useful in many situations of interest as we show by the examples of trees and antitrees. Another way to estimate isoperimetric constants is provided by the volume growth. Namely, we can apply the exponential volume growth estimates for regular Dirichlet forms from [56] (see also [32], [47]) to prove upper bounds (Brooks-type estimates [7]) for quantum graphs (see Theorem 7.1). However, this can be done under the additional assumption that the metric graph is complete with respect to the natural path metric (notice that in this case H 0 is essentially self-adjoint and H coincides with its closure, see Corollary 2.3).
The quantities λ 0 (H) and λ ess 0 (H) are of fundamental importance for several reasons. From the spectral theory point of view, the positivity of λ 0 (H) or λ ess 0 (H) corresponds to, bounded invertibility or Fredholmness of the operator H. Moreover, λ ess 0 (H) = +∞ holds precisely when the spectrum of H is purely discrete, which is further equivalent to the compactness of the embedding H 1 (G) into L 2 (G). It is difficult to overestimate the importance of λ 0 (H) and λ ess 0 (H) in applications. For example, in the theory of parabolic equations λ 0 (H) gives the speed of convergence of the system towards equilibrium. On the other hand, Cheeger-type inequalities have a venerable history. Starting from the seminal work of J. Cheeger [15], where a connection between the isoperimetric constant of a compact manifold and a first nontrivial eigenvalue of the Laplace-Beltrami operator was found, this topic became an active area of research in both manifolds and graphs settings. One of the most fruitful applications of Cheeger's inequality in graph theory (this inequality was first proved independently in [19,21] and [1,2]) is in the study of networks connectivity, namely, in constructing expanders (see [16,18,33,43]). Let us also mention that the positivity of the isoperimetric constant (also known as a strong isoperimetric inequality) is of fundamental importance in the study of random walks on graphs (we refer to [58] for further details).
Let us now finish the introduction by describing the content of the article. First of all, we review necessary notions and facts on infinite quantum graphs in Section 2, where we introduce the pre-minimal operator H 0 (Section 2.2), discuss its essential self-adjointness (Section 2.3) and the corresponding quadratic form t G (Section 2.4), and also touch upon its connection with the difference Laplacian (1.1) (Section 2.5).
Section 3 contains our first main result, Theorem 3.4, which provides the Cheegertype estimate for quantum graphs. Its proof follows closely the line of arguments as in the manifold case with the only exception, Lemma 3.7, which enables us to replace the isoperimetric constant from [46] (see also [38,51]) by the quantity (3.3) having a combinatorial structure. The latter also reveals connections with the combinatorial isoperimetric constant α comb from [2,19], which measures connectedness of the underlying combinatorial graph, and with the discrete isoperimetric constant α d introduced recently in [5] for the difference Laplacian (1.1). Bearing in mind the importance of both α comb and α d in applications as well as the fact that these quantities are widely studied, we discuss these connections in Sections 4.
Similar to manifolds and combinatorial Laplacians, one can estimate λ 0 (H) and λ ess 0 (H) by using the isoperimetric constant not only from below but also from above (Lemma 5.1). However, the price we have to pay is the existence of a positive lower bound on the edges lengths, inf e∈E |e| > 0. Combining these estimates with the results from Section 4, we conclude that in this case the positivity of λ 0 (H) (resp., λ ess 0 (H)) is equivalent to the validity of a strong isoperimetric inequality, i.e., α comb > 0 (resp., α ess comb > 0). In Section 6, we introduce a quantity which may be interpreted as a curvature of a metric graph. Firstly, using this quantity we are able to obtain estimates on the isoperimetric constant. Secondly, we discuss its connection with the curvatures introduced for combinatorial Laplacians in [20] and for unbounded difference Laplacians in [5]. The latter, in particular, enables us to obtain simple discreteness criteria for σ(H) (see Lemma 6.5 and Corollary 6.6), which to a certain extent can be seen as the analogs of discreteness criteria from [22] and [29].
The estimates in terms of the volume growth are given in Section 7. In the final section, we consider several illustrative examples. The case of trees is treated in Section 8.1. We show that for trees without inessential vertices and loose ends (vertices having degree 1), λ 0 (H) > 0 if and only if sup e |e| < ∞. Moreover, the spectrum of H is purely discrete if and only if the number #{e ∈ E : |e| > ε} is finite for every ε > 0. Notice that under the additional symmetry assumption that a given metric tree is regular these results were observed by M. Solomyak [55]. The case of antitrees is considered in Section 8.2. We provide some general estimates and also focus on two particular examples of exponentially and polynomially growing antitrees. In particular, it turns out that for a polynomially growing antitree, our results provide rather good estimates for λ 0 (H) and λ ess 0 (H) (see Example 8.9). In the last subsection, we consider the case of Cayley graphs of finitely generated groups. Similar to combinatorial Laplacians, the amenability/non-amenability of the underlying group plays a crucial role.

Combinatorial and metric graphs.
In what follows, G d = (V, E) will be an unoriented graph with countably infinite sets of vertices V and edges E. For two vertices u, v ∈ V we shall write u ∼ v if there is an edge e u,v ∈ E connecting u with v. For every v ∈ V, we denote the set of edges incident to the vertex v by E v and is called the degree (or combinatorial degree) of a vertex v ∈ V. When there is no risk of confusion which graph is involved, we shall write deg instead of deg G . By #(S) we denote the cardinality of a given set S.
We shall always make the following assumption.
, connected (for any two vertices u, v ∈ V there is a path connecting u and v), and simple (there are no loops and multiple edges).
Next we assign each edge e ∈ E a finite length |e| ∈ (0, ∞). In this case G := (V, E, | · |) = (G d , | · |) is called a metric graph. A metric graph G can be considered as a one-dimensional complex. By assigning each edge a direction and calling one of its vertices the initial vertex e 0 and the other one the terminal vertex e i , every edge e ∈ E can be identified with the interval I e = (0, |e|). As the name suggests, G can be equipped with the natural path metric ̺ 0 (the distance between two points x, y ∈ G is defined as the length of the "shortest" path connecting x and y) and this metric enables one to interpret G as a topological space, that is, the union of all edges glued together at certain endpoints (see, e.g., [10,Chapter 1.3]).
Also throughout this paper we shall assume the following conditions. In fact, Hypothesis 2.2 is not a restriction for our purposes (see Lemma 2.9 and also Remark 2.10(i)).
This assumption is not a restriction at all since vertices of degree 2 are irrelevant for the spectral properties of the Kirchhoff Laplacian and hence can be removed (see, e.g., [38]).

2.2.
Kirchhoff 's Laplacian. Given a metric graph G satisfying Hypothesis 2.1-2.3, let us introduce the Hilbert space L 2 (G) of functions f : G → C such that The subspace of compactly supported L 2 (G) functions will be denoted by (2.7) Integrating by parts one obtains that H 0 is symmetric. We call its closure the minimal Kirchhoff Laplacian. Notice that the values of f at the vertices (2.4) and one-sided derivatives (2.5) do not depend on the choice of orientation on G. Moreover, the second derivative is also independent of orientation on G and hence so is the operator H 0 . and we shall consider the operator H 0 with mixed boundary conditions (either Neumann or Dirichlet) at the vertices v ∈ V of the graph G such that deg(v) = 1.
In the rest of our paper, we shall denote by V D (respectively, by V N ) the set of vertices v ∈ V such that deg(v) = 1 and the Dirichlet condition (2.9) (respectively, the Neumann condition (2.9)) is imposed at v. The sets of corresponding edges will be denoted by E D and E N , respectively.
2.3. Self-adjointness. In the rest of our paper we shall always assume that the graph G d is infinite, that is, both sets V and E are infinite (since G d is assumed to be locally finite). In this case the operator H 0 is not necessarily essentially self-adjoint (that is, its closure may have nonzero deficiency indices) and finding self-adjointness criteria is a challenging open problem. The next results were proved recently in [24]. Define the weight function m : V → R >0 by and then let p m : E → R >0 be given by The path metric ̺ m on V generated by p m is defined by where the infimum is taken over all paths connecting u and v. Replacing p m in (2.12) by the edge length | · |, we end up with the natural path metric ̺ 0 on V. Clearly, (V, ̺ m ) is complete if so is (V, ̺ 0 ) and hence we arrive at the following Gaffney-type theorem for quantum graphs.

Corollary 2.3 ([24]
). If G equipped with a natural path metric is complete as a metric space, then H 0 is essentially self-adjoint.
The next well known result (see [10,Theorem 1.4.19]) also immediately follows from Theorem 2.2.
then H 0 is essentially self-adjoint.

Quadratic forms.
In this section we present the variational definition of the Kirchhoff Laplacian. Consider the quadratic form For every f ∈ dom(H 0 ), an integration by parts gives Clearly, the form t 0 G is nonnegative. Moreover, it is closable since H 0 is symmetric. Let us denote its closure by t G and the corresponding domain by H 1 0 (G) := dom(t G ). By the first representation theorem, there is a unique nonnegative self-adjoint operator corresponding to the form t G .
Definition 2.5. The self-adjoint nonnegative operator H associated with the form t G in L 2 (G) will be called the Kirchhoff Laplacian.
If the pre-minimal operator H 0 is essentially self-adjoint, then H coincides with its closure. In the case when H 0 is a symmetric operator with nontrivial deficiency indices, the operator H is the Friedrichs extension of H 0 . Remark 2.6. Of course, one can define the operator H as the operator corresponding to the maximally defined form where dom(t (N ) coincide if and only if H 0 is essentially self-adjoint. We are not aware of a description of the self-adjoint operator H N associated with the form t (N ) G if the pre-minimal operator has nontrivial deficiency indices. Moreover, to the best of our knowledge, the description of deficiency indices of H 0 is still a widely open problem.
Remark 2.7. If at some vertices v ∈ V with deg(v) = 1 the Neumann condition (2.8) is replaced by the Dirichlet condition (2.9), then the corresponding form domain will be denoted by H 1 0 (G). Notice that The bottom of the spectrum of H can be found by using the Rayleigh quotient . (2.20) Moreover, the bottom of the essential spectrum is given by where the sup is taken over all finite subgraphs G of G. Here for any G ⊂ G we define H 1 0 (G \ G) as the set of H 1 0 (G \ G) functions satisfying the following boundary conditions: for vertices in G \ G having one or more edges in G, we change the boundary conditions from Kirchhoff-Neumann to Dirichlet; for all other vertices in G \ G, we leave them the same. This equality is known as a Persson-type theorem (or Glazman's decomposition principle in the Russian literature, see [31]) and its proof in the case of quantum graphs is analogous to the case of Schrödinger operators (see, e.g., [17,Theorem 3.12]). Remark 2.8. Let us mention that the following equivalence holds true The implication " ⇐ " is obvious. However, λ 0 (H) = 0 and λ ess 0 (H) = 0 holds only if 0 is an isolated eigenvalue. On the other hand, (2.16) implies that 0 is an This immediately implies that the equivalence (2.22) holds true whenever mes(G) = ∞. On the other hand, it turns out that ½ / ∈ H 1 0 (G) if mes(G) < ∞ and hence 0 is never an eigenvalue of H (cf. Corollary 3.5). Actually, it does not seem trivial to prove this fact. In particular, the latter implies that t G = t (N ) G if the metric graph G has finite total volume, mes(G) < ∞. The analysis of this case is postponed to a separate publication.
by setting it zero on remaining edges. Thus, .
Let K G be the set of all finite, connected subgraphs of G ordered by the inclusion relation "⊆" and hence K G is a net. Moreover, where the limit is understood in the sense of nets and in this case we will say that G tends to G. The next result provides an estimate, which easily follows from (2.20)-(2.21).
where the infimum is taken over all finite subsets E of E. Then is piecewise linear on G (that is, it is linear on every edge e ∈ E) and f 0 ∈ H 1 0 (G) takes zero values at the vertices V. It is straightforward to check that Now the estimates (2.25) and (2.24) easily follow from the decomposition (2.26).
and then taking into account (2.20) and (2.21), we arrive at (2.25). (ii) The second inequality in (2.25) implies that (2.24) is necessary for the spectrum of H to be purely discrete. Notice that ℓ * ess (G) = 0 means that the number #{e ∈ E| |e| > ε} is finite for every ε > 0.
(iii) The estimates (2.25) can be slightly improved by noting that we can use other test functions on the edges e ∈ E N to improve the bound (π/|e|) 2 by (π/2|e|) 2 . For example, we get the following estimate (2.28) 2.5. Connection with the difference Laplacian. In this section we restrict for simplicity to the case of Neumann boundary conditions at the loose ends, that is, Let the weight function m : V → R >0 be given by (2.10). Consider the difference Laplacian defined in ℓ 2 (V; m) by the expression The operator h 0 is a nonnegative symmetric operator. Denote its Friedrichs extension by h.
It was observed in [24] that the operators H and h are closely connected (for instance, by [24, Corollary 4.1(i)], H 0 and h 0 are essentially self-adjoint only simultaneously). In fact, it is not difficult to notice a connection between H and h by considering their quadratic forms (see [24,Remark 3.7]). Namely, let L = ker(H max ) be the kernel of H max , which consists of piecewise linear functions on G. Every f ∈ L can be identified with its values {f (e i ), f (e o )} e∈E on V and, moreover, Now restrict ourselves to the subspace L cont = L ∩ C c (G). Clearly, defines an equivalent norm on L cont since the Cauchy-Schwarz inequality immediately implies On the other hand, for every f ∈ L cont we get (2.33) Hence we end up with the following estimate.
In this particular case spectral relations between H and h have already been observed by many authors (see [6], [ Hence for equilateral graphs we obtain The latter together with (2.34) imply that for equilateral graphs the following equivalence holds true Remark 2.14. Since 4 for all x ∈ [0, π 2 /4], (2.36) implies the following estimate for equilateral quantum graphs which improves (2.34). Moreover, the constant π 2 /4 is sharp in the equilateral case. However, it remains unclear to us how sharp is the estimate (2.34).

The Cheeger-type bound
For every G ∈ K G we define the boundary of G with respect to the graph G as the set of all vertices For a given finite subgraph G ⊂ G we then set Remark 3.1. Let us stress that our definition of a boundary is different from the combinatorial one. In particular, we define the boundary as the set of vertices whereas the combinatorial definition counts the number of edges connecting V with its complement V \ V.
Recall that for any G ∈ K G we consider G \ G with the following boundary conditions: for vertices in G \ G having one or more edges in G, we change the boundary conditions from Kirchhoff-Neumann to Dirichlet; for all other vertices in G \ G, we leave them the same. These boundary conditions imply that for a subgraph Y ∈ K G\ G , where the left-hand side is the boundary of Y with respect to G \ G (with the new Dirichlet conditions) and the right-hand side is the boundary with respect to the original graph G. Hence, Choosing G as an edge e ∈ E or a star E v with some v ∈ V, one gets the following simple bounds on the isoperimetric constant The next result is the analog of the famous Cheeger estimate for Laplacians on manifolds [15].
As an immediate corollary we get the following result.
(iv) If the metric graph G has finite total volume, mes(G) < ∞, then H is a uniformly positive operator with purely discrete spectrum.
Proof. Clearly, we only need to prove (iv). Since mes(G) < ∞ and taking (3.3) into account, we immediately obtain which together with (3.8) implies the inequality λ 0 (H) > 0. Next, using (3.4) together with the estimate (3.9) and the net property of K G , one gets α ess (G) = ∞, which finishes the proof.
Before proving the estimates (3.8) we need several preliminary lemmas. For every measurable function h : G → R and every t ∈ R let us define the set (3.10) The next statement is known as the co-area formula and we give its proof for the sake of completeness. can be written as M e = n I n for (at most countably many) disjoint open intervals I n ⊆ (0, |e|). Since h is strictly monotone on each of these intervals, Here mes(X) denotes the Lebesgue measure of X ⊆ R. Moreover, by continuity of h, it is straightforward to check that ½ h(In) (t) = #(∂Ω h (t) ∩ I n ) for all t ∈ R.
Hence we end up with is the set of critical points of h. By Sard's Theorem [53], h(M c e ) has Lebesgue measure zero and hence Assume now that h : G → R is an arbitrary function satisfying the assumptions. Then we get Since V is countable, we arrive at (3.11).
Next it will turn out useful to rewrite the Cheeger constant (3.3) in the following way. Let where U G = ∪ G∈KG U G and Here ∂U is the boundary of the set U in the sense of the natural topology on G. Indeed, assume first that G ∈ K G and identify G with a closed subset of the graph.
For a sufficiently small ε > 0, we cut out a ball B ε (v) of radius ε at each point in v ∈ ∂ G G and obtain the set We have U ∈ U G and, moreover, ∂U has precisely deg(∂ G G) points. In total, Letting ε tend to zero, we obtain the desired inequality.
(ii) To prove the other inequality, let U ∈ U G and G = ( V, E) be the finite subgraph consisting of all edges e ∈ E with e ∩ U = ∅ and all vertices incident to such an edge. Clearly, mes(U ) ≤ mes( G). Also, by (3.2), Since U is open, every point of ∂ G G is not in U . Therefore, every edge in the subgraph G connected to a vertex in ∂ G G must contain at least one boundary point of U . If both vertices of the edge are in ∂ G G, it must even contain at least two boundary points of U . Also, since V ∩ ∂U = ∅, the boundary points lie in the strict interior of each edge and therefore cannot coincide for different edges. Thus, deg(∂ G G) ≤ #(∂U ).
Finally, notice that G might be disconnected. If it is the case, then write G = ∪ n G n as a disjoint, finite union of connected subgraphs G n ∈ K G . Then which implies that α(G) ≥ α(G).
Now we are in position to prove the Cheeger-type estimates (3.8).
Proof of Theorem 3.4. Let us show that the following inequality holds true for all f ∈ dom(t 0 G ) = dom(H 0 ). Without loss of generality we can restrict ourselves to real-valued functions. So, suppose f ∈ dom(H 0 ) is real-valued. Observe that (see, e.g., [30,Lemma I.4.1]) Next we want to use Lemma 3.
then t ∈ f 2 (V) by continuity of f 2 . Since V and hence f 2 (V) are countable, we get that Ω f 2 (t) ∈ U G for almost every t > 0. Thus, in view of Lemma 3.7 On the other hand, applying Lemma 3.6 to h = f 2 and then the Cauchy-Schwarz inequality, we get Combining the last two inequalities, we arrive at (3.15), which together with the Rayleigh quotient (2.20) proves the first inequality in (3.8).
The proof of the second inequality in (3.8) follows the same line of reasoning since by (2.21) , for every finite subgraph G of G. Notice that the boundary conditions on G \ G are defined after (3.4).
Remark 3.8. The Cheeger estimate for finite quantum graphs was first proved in [46] (see also [51, §6] and [37]). Our result extends [46, Theorem 3.2] to the case of infinite graphs and also provides a bound on the essential spectrum of H. Moreover, our definition of the isoperimetric constant (3.8) is different from the one given in [46]. Namely, (3.8) is purely combinatorial since the infimum is taken over finite connected subgraphs of G, although the definition in [46] (see also [38,51]) is similar to (3.12).
Taking into account the equivalence (2.22), let us finish this section with the next observation. Proof. Clearly, we only need to prove the implication α(G) = 0 ⇒ α ess (G) = 0. Assume the converse, that is, there is an infinite graph G satisfying Hypotheses 2.1-2.3 such that α(G) = 0 and α ess (G) > 0. Then by (3.3), there is a sequence {G n } ⊂ K G such that On the other hand, (3.4) implies that there is G ∈ K G such that α(G \ G) = α 0 > 0. In particular, taking into account (3.5), the latter is equivalent to the fact that which leads to a contradiction. Indeed, by construction, lim n→∞ mes(G n ) = ∞ and hence mes(G n \ G) = mes(G n )(1 + o(1)) as n → ∞. It remains to note that

Connections with discrete isoperimetric constants
For every vertex set X ⊆ V, we define its boundary and interior edges by where m : V → (0, ∞) is defined by (2.10) (in fact, m(v) = mes(E v ) for every v ∈ V). Following [5] (see also [24 The discrete isoperimetric constant of the graph G is then given by Moreover, we need the discrete isoperimetric constant at infinity Since ̺ 0 is intrinsic in the sense of [5,27], we arrive at the following Cheeger-type estimates for the discrete Laplacian (2.29)-(2.30) (see [

5, Theorems 3.1 and 3.3])
The next result provides a connection between isoperimetric constants.
In particular, the isoperimetric constants at infinity (3.4) and (4.3) satisfy Proof. (i) First, let X ⊂ V be finite. Let also G = ( V, E) be the finite subgraph of G consisting of all edges with at least one vertex in the set X. Observe that Note that for every v ∈ X, the whole star E v attached to it is in G. Therefore, every vertex from ∂ G G is not in X. Now consider an edge e in the subgraph G which is connected to a vertex v ∈ ∂ G G. Then its other endpoint must be in X (because of the definition of G). Hence #{e| e connects v and X} ≤ #{e ∈ E| e connects X and V\X} = #(E b (X)).
Splitting G in finitely many connected components as in the proof of Lemma 3.7, we arrive at the first inequality in (4.5).
To prove the second inequality, assume G ∈ K G . Write E = E 0 ∪ E 1 ∪ E 2 , where E 0 , E 1 , E 2 are the sets of edges in the subgraph with, respectively, none, one, and two vertices in ∂ G G. Clearly, (4.7) Now define the finite vertex set X := V\∂ G G. We have (ii) To prove (4.6), let first X ⊆ V be a finite and connected (in the sense that for two vertices in X, there always exists a path connecting them and only passing through vertices in X) set of vertices. Then the subgraph G X ⊆ G consisting of all edges with both vertices in X is finite and connected. Now note that for a finite vertex set Y ⊆ V \ X, the subgraph G Y defined above is contained in G \ G X . Hence taking into account (3.5) and using the same line of reasoning as in (i), we get α(G \ G X ) ≤ 2α d (V \ X). Finally, choose an increasing sequence {X n } ⊆ V of finite and connected vertex sets such that every finite vertex set X ⊆ V is eventually contained in X n . Then the corresponding sequence { G n } ⊆ K G of subgraphs is increasing and every finite, connected subgraph G ∈ K G is eventually contained in G n . In view of (3.6), we obtain the first inequality in (4.6) by taking limits.
To prove the second, for a subgraph G 0 ∈ K G , choose X to be the set of vertices in G 0 . Let G ∈ K G\G0 . If a vertex v is both in V and in X, then it has at least one incident edge which lies in the cut out graph G 0 and therefore v ∈ ∂ G G. Thus, the vertex set Y = V\∂ G G satisfies Y ∩ X = ∅. Refining the previous estimate, Choosing an increasing sequence {G n } ⊆ K G such that every G 0 ∈ K G is eventually contained in G n and applying the same limit argument as before, we arrive at the second inequality in (4.6). Combining (4.5) with Corollary 3.5, we obtain Theorem 4.18 from [24].
The spectrum of H is purely discrete if the number #{e ∈ E : |e| > ε} is finite for every ε > 0 and α ess d (V) = ∞. Proof. We only need to mention that ℓ * ess (G) = 0 if and only if the number #{e ∈ E : |e| > ε} is finite for every ε > 0. Moreover, in this case it follows from (4.6) that α ess (G) = α ess d (V).
Finally, let us mention that in the case of equilateral graphs the discrete isoperimetric constants coincide with the combinatorial isoperimetric constants introduced in [21]: Comparing (4.8) with (4.2) and (4.3) and noting that for all v ∈ V, one easily derives the estimates . Here and the supremum is taken over all finite subsets E of E. Moreover, taking into account Lemma 4.1, we get the following connection between our isoperimetric constants and the combinatorial ones: .

Upper bounds via the isoperimetric constant
It is possible to use the isoperimetric constants to estimate λ 0 (H) and λ ess 0 (H) from above, however, for this we need to impose additional restrictions on the metric graph.
α ess (G). Assume a subgraph G 0 ∈ K G and a finite, connected subgraph G = ( V, E) of G \ G 0 . Then define g ∈ H 1 c (G \ G 0 ) by setting where E 0 , E 1 , E 2 are defined as in the previous subsection and |x e − y| denotes the distance between x e ∈ e and some y ∈ e. If G 0 = ∅ and v ∈ G \ G 0 is a vertex with at least one incident edge in G 0 , then either v is not in V or v is a boundary vertex of G. In both cases, g vanishes at v. Therefore, g ∈ H 1 (G \ G 0 ). Next we get and, in view of (4.7), Choosing φ(x) = √ 2 sin( π 2 x), we obtain the estimate g ′ 2 Choosing G 0 = ∅, (2.20) and (3.8) imply the first inequality in (5.1). Now assume Finally, using (2.23) and (3.6) we end up with Combining Lemma 5.1 with the Cheeger-type bounds (3.8) and the estimates (4.10)-(4.11) and taking into account Lemma 3.9, we immediately get the following result.
Corollary 5.2. If ℓ * (G) > 0 and ℓ * (G) < ∞, then the following are equivalent: (ii) Notice that (5.1) is better than (2.25) only if the isoperimetric constant satisfies [12], Buser noticed that the isoperimetric constant can be used for obtaining upper estimates on the spectral gap for Laplacians on compact Riemannian manifolds. Hence estimates of the type (5.1) are often called Buser-type estimates. Let us mention that for combinatorial Laplacians a Buser-type estimate was first proved in [2] (see also [16,18]). For finite quantum graphs, a Buser-type bound can be found in [38,Proposition 0.3], which is, however, different from our estimate (5.1).

Bounds by curvature
Despite the combinatorial nature of isoperimetric constants (3.3) and (3.4), it is known that computation of the combinatorial isoperimetric constant (4.8) is an NP-hard problem (see [33,35] for further details). Our next aim is to introduce a quantity, which provides estimates for α(G) and α ess (G) and also turns out to be very useful in many situations (see Section 8).
Suppose now that our graph is oriented, that is, every edge is assigned a direction. For every v ∈ V, let E + v and E − v be the sets of outgoing and incoming edges, respectively. Next define the function K : Note that K can take both positive and negative values, and K(v) = −∞ whenever #(E + v ) = ∅. Lemma 6.1. Assume G is an oriented graph such that the function K is positive. Then the isoperimetric constant (3.3) satisfies Proof. Let G ∈ K G be a finite and connected subgraph. For every v ∈ V, denote by E + v ( G) and E − v ( G) the sets of outgoing and incoming edges in G. Since K(v) > 0 is positive, we get , for all v ∈ V. Therefore, .

Moreover, for any non-boundary point
Combining this with the previous estimates, we end up with the following bound which proves the claim.
Remark 6.2. The function K is sometimes interpreted as curvature. Several notions of curvature have been introduced for discrete and combinatorial Laplacians. Perhaps, the closest one to (6.1) have been introduced in [36]. Namely, since the natural path metric ̺ 0 is intrinsic, define the function K d : V → R by .

4)
and hence in this case 2 , v ∈ V. (6.5) It seems there is no nice connection between K and K d in the general case.
Remark 6.3. Let us also mention that Lemma 6.1 can be seen as the analog of [5, Theorem 6.2], where the following bound for the discrete isoperimetric constant was established: if K d is nonnegative on V. Combining (6.6) with the second inequality in (4.5), we end up with the following bound 2 α(G) In what follows we shall call the function K comb : V → Q ∪ {−∞} defined by (6.4) as the combinatorial curvature (in [20, p. 32], K d is called a curvature of the combinatorial distance spheres). Note that the curvature can take both positive and negative values, and K comb (v) = −∞ whenever #(E + v ) = ∅. The next simple estimate turns out to be very useful in applications. Lemma 6.4. Assume K comb is positive on V and Then the isoperimetric constant (3.3) satisfies Proof. Noting that K comb is positive and comparing (6.4) with (6.1), we get for all v ∈ V. Hence the claim follows from Lemma 6.1.
With a little extra effort and using an argument similar to that in the proof of (4.5) one can show the following bounds. and Combining Lemma 6.5 with the Cheeger-type estimate, we immediately get the following result. Corollary 6.6. If G is an oriented graph such that the function K comb is nonnegative on V, then
Remark 6.7. Let us mention that in the case when K ess comb (V) = 0 the condition ℓ * ess (G) = 0 is no longer sufficient for the discreteness. For further details we refer to Section 8.2 and, more specifically, to the example of polynomially growing antitrees (see Example 8.7).

Growth volume estimates
Here we plan to exploit the results from [56] to get upper bounds on the spectra of quantum graphs in terms of the exponential volume growth rates, the so-called Brooks-type estimates (cf. [7], [32], [56] for further details and references). Following [56], we introduce the following notation. For every x ∈ G and r > 0, let B r (x) := {y ∈ G| ̺ 0 (x, y) < r}. Notice that µ x (G) does not depend on x ∈ G if G = ∪ r>0 B r (x) for some (and hence for all) x ∈ G. If both conditions are satisfied, then we shall write µ(G) instead of µ x (G).
Proof. The first and the last inequalities in (7.5) are obvious and hence it remains to show that λ ess 0 (H) ≤ Notice that by Corollary 2.3, the pre-minimal operator H 0 is essentially self-adjoint and hence H is its closure. Let us consider the corresponding quadratic form t G defined as the closure in L 2 (G) of the form t 0 G (see (2.15) and (2.16)). It is not difficult to check that the form t G is a strongly local regular Dirichlet form (see [28] for definitions). On the other hand, using the Hopf-Rinow type theorem for graphs (see [34]), with a little work one can show that every ball B r (x) is relatively compact if (V, ̺ 0 ) is complete. Therefore, by [56,Theorem 5] Noting that mes(B 1 (x)) ≥ 1 for all x ∈ G and taking into account [32, Remark (e) on p.885], we arrive at the desired estimate.
The next result is straightforward from Theorem 7.1.
Corollary 7.2. Let (V, ̺ 0 ) be complete as a metric space. Then: Remark 7.3. Clearly, to compute or estimate µ * (G) is a much more involved problem comparing to that of µ(G). However, it might happen that µ * (G) < µ(G) and hence µ * (G) provides a better bound (see Example 8.4).
Remark 7.4. Let us mention that these results have several further consequences for the heat semigroup e −tH generated by the operator H. For example, µ * (G) = 0 implies the exponential instability of the corresponding heat semigroup on L p (G) for all p ∈ [1, ∞] (see [56,Corollary 2]).
We finish this section with comparing the estimates (7.5) with the ones obtained in [24] in terms of the volume growth of the corresponding discrete graph. Following [32] (see also [24, §4.3]), define the constant Here Lemma 7.5. If ℓ * (G) < ∞ and (V, ̺ 0 ) is complete as a metric space, then Proof. First observe that for all v ∈ V and r > 0, which immediately implies µ(G) ≤ µ d (G). Similarly, we also get m(B r (v)) ≤ 2mes(B r+ℓ * (v)) (7.8) for all v ∈ V and r > 0 and hence which finishes the proof of (7.7).
Remark 7.6. A few remarks are in order.
(i) On the one hand, it does not look too surprising that the exponential growth rates for two Dirichlet forms t G and t h coincide. In particular this reflects the equivalence (2.37) in the case of sub-exponential growth rates. However, comparing (7.7) with the fact that there is no equality between λ 0 (H) and λ 0 (h) (see Section 2.5), one can conclude that in the case of an exponential growth of volume balls, (7.5) might not lead to qualified estimates (and examples of trees and antitrees in the next section confirm this observation). (ii) Combining (7.7) with Corollary 7.2 we obtain Theorem 4.19 from [24].

Examples
In this section we are going to apply our results to certain classes of graphs (trees, antitrees, and Cayley graphs of finitely generated groups). Let us also recall that we always assume Hypotheses 2.1-2.3 to be satisfied. 8.1. Trees. Let us first recall some basic notions. A connected graph without cycles is called a tree. We shall denote trees (both combinatorial and metric) by T . Notice that for any two vertices u, v on a tree T = (V, E) there is exactly one path P connecting u and v. A tree T = (V, E) with a distinguished vertex o ∈ V is called a rooted tree and o is called the root. In a rooted tree the vertices can be ordered according to (combinatorial) spheres. Namely, let d(·) := d(o, ·) be the combinatorial distance to the root o and S n be the n-th (combinatorial) sphere, i.e., the set of vertices v ∈ V with d(v) = n. A vertex in the (n + 1)-th sphere, which is connected to v in the n-th sphere, is called a forward neighbor of v. In what follows, we define an orientation on a rooted tree according to combinatorial spheres, that is, for every edge e its initial vertex belongs to the smaller combinatorial sphere.
We begin with the following simple estimate for rooted trees. According to the choice of orientation, we get K comb (o) = deg(o) and Hence we easily get and therefore we end up with the following estimate.
Lemma 8.1. Assume T is a rooted tree without loose ends. Then In particular, λ 0 (H) > 0 if and only if ℓ * (G) < ∞ and the spectrum of H is purely discrete if and only if ℓ * ess (G) = 0. Proof. The proof immediately follows from Corollary 6.6, Remark 2.10(i) and the fact that the combinatorial curvature admits the following bound (take also into account Hypothesis 2.3) In the case of trees the estimates (8.1) can be improved, however, instead of providing these generalizations we are going to consider only one particular case. Figure 1. Tree with loose ends.
Example 8.3 (Bethe lattices). Fix β ∈ Z ≥3 and consider the combinatorial graph, which is a rooted tree such that all vertices have degree β. This type of trees is called Bethe lattices (also known as Cayley trees or homogeneous trees) and they will be denoted by T β . Suppose that the corresponding metric graph is equilateral, that is, |e| = 1 for all e ∈ E. By abusing the notation, we shall denote the corresponding metric graph by T β too. Then one computes Noting that K β ∈ [1/2, 1) and applying Lemma 8.1, we arrive at the following estimate On the other hand, it is straightforward to check that (see, e.g., [20]) In particular, this implies that the equality holds in the second inequality in (4.5). Moreover, the spectra of both operators H and h can be computed explicitly (see, e.g., [55,Example 6.3] or [20, Theorem 1.14] together with Theorem 2.12) and, in particular, Comparing the last equality with the estimate (8.2), one can notice a gap between these estimates. Let us mention that and thus the volume growth estimates (7.5) do not provide a reasonable upper bound for large values of β. ♦ Finally, we would like to mention that the absence of loose ends in Lemma 8.1 is essential as the next example shows.
Example 8.4 (A "sparse" tree with loose ends). Consider the half-line R ≥0 as an equilateral graph with vertices at the integers. Let us write v n for the vertex placed at n ∈ Z ≥0 . Then, we will modify this graph by attaching edges to the vertices v n with n ≥ 1. More precisely, to the j 2 -th vertex v j 2 with j ∈ Z ≥1 , we attach 2 j 2 edges and to every other vertex v n with n / ∈ {j 2 } j≥1 , we attach exactly one edge (see Figure 1). Figure 2. Example of an antitree with s n = n + 1.
It is interesting to mention that in this case µ(T ) = log(2) > 0. Indeed, and hence for all r > 1 we get which implies the desired equality. ♦

8.2.
Antitrees. Let G d = (V, E) be a connected combinatorial graph. Fix a root vertex o ∈ V and then order the graph with respect to the combinatorial spheres S n , n ∈ Z ≥0 (notice that S 0 = {o}). The connected graph G d is called an antitree if every vertex in S n is connected to every vertex in S n+1 and there are no horizontal edges, i.e., there are no edges with all endpoints in the same sphere (see Figure 2). Clearly, an antitree is uniquely determined by the sequence s n := #(S n ), n ∈ Z ≥1 . Let us denote antitrees by the letter A and also define the edge orientation according to the combinatorial ordering, that is, for every edge e its initial edge is the one in the smaller combinatorial sphere. It turns out that the curvatures of antitrees can be computed explicitly. Namely, define the following quantities: and K 0 := 1, K n+1 := 1 − s n s n+2 (8.5) for all n ∈ Z ≥0 .
Lemma 8.5. If A is an antitree, then and Proof. The proof follows by a direct inspection since K comb (v) = K n for all v ∈ S n and n ∈ Z ≥0 .
Combining Lemma 8.5 with the estimates for the corresponding isoperimetric constants (e.g., Corollary 6.6), we immediately end up with the estimates for λ 0 (H) and λ ess 0 (H). Let us demonstrate this by considering two examples. Example 8.6 (Exponentially growing antitrees). Fix β ∈ Z ≥2 and let A β be an antitree with sphere numbers s n = β n . Then K 0 = 1 and for all n ∈ Z ≥1 . Hence by Lemma 8.5 .
Applying Lemmas 6.1 and 6.5 together with Theorem 3.4 and Lemma 2.9, we get and (8.10) In particular, these bounds imply that the Kirchhoff Laplacian H β is uniformly positive if and only if ℓ * (A β ) < ∞. Moreover, its spectrum is purely discrete exactly when ℓ * ess (A β ) = 0 (cf. Corollary 6.6). Finally, let us compare these estimates with the volume growth estimates under the assumption that the tree is equilateral. In this case, On the other hand, and then (7.4) implies that µ(A β ) = 2 log(β). With a little more work one can show that µ * (A β ) = µ(A β ) = 2 log(β).
Thus, we obtain vol * (r) = inf x∈G mes(B r (x)) mes(B 1 (x)) ≥ inf n≥0 β 2n+1 β 2⌊r⌋ −1 which shows that µ * (A β ) ≥ 2 log(β) and hence we are done. Notice that the volume growth estimates (7.5) do not provide a reasonable upper bound for large values of β. ♦ Example 8.7 (Polynomially growing antitrees). Fix q ∈ Z >0 and let A q be the antitree with sphere numbers s n = (n + 1) q , n ≥ 0 (the case q = 1 is depicted on Figure 2). Then K n = 1 − n q (n + 2) q = 1 − Clearly, further analysis heavily depends on the behavior of the sequence {ℓ n }. Let us consider one particular case. Fix an s ≥ 0 and assume now that |e| = (n + 1) −s for each edge e connecting S n and S n+1 . Let us denote the corresponding Kirchhoff Laplacian by H q,s . It is not difficult to show by applying Theorem 2.2 that the corresponding pre-minimal operator is essentially self-adjoint whenever s ≤ q + 1, however, (V q , ̺ 0 ) is complete exactly when s ∈ [0, 1].
Remark 8.8. Let us stress that it would be interesting to understand for which parameters s and q the pre-minimal operator H 0 is essentially self-adjoint if s > q +1. The symmetry of the underlying combinatorial antitree suggests that the deficiency indices are either zero or infinity. The description of self-adjoint extensions in the latter case also seems to be a rather nontrivial and interesting problem. In the case s = 1, it is easy to show that the sequence {K n /ℓ n } is strictly increasing and hence this is also true for all s > 1. Hence K(A q ) = K(o) = 1, s ≥ 1.
Moreover, the corresponding isoperimetric constant is given by α(A q ) = K(A q ) = 1 (to see this just take the ball B 1 (o) as a subgraph G and then one gets α(A q ) ≤ 1, which together with (6.2) implies the equality).
Next let us compute µ(A q ) assuming that s ∈ [0, 1] (otherwise we can't apply the result from Section 7). Set  if and only if s ∈ [0, 1). If s ≥ 1, then the operator H q,s is uniformly positive and 1 4 ≤ λ 0 (H q,s ) ≤ π 2 , λ ess 0 (H q,s ) = q 2 , s = 1, +∞, s > 1. (8.15) Remark 8.10. The exact value of λ 0 (H q,s ) for s ≥ 1 or at least its asymptotic behavior with respect to q remains an open problem. ♦ 8.3. Cayley graphs. Suppose Γ is a finitely generated (infinite) group with the set of generators S. The Cayley graph C(Γ, S) of Γ with respect to S is the vertex set Γ and u ∼ v exactly when u −1 v ∈ S. This graph is connected, locally finite and regular (deg(v) = #S for all v ∈ Γ). We assume that the unit element o does not belong to the set S (this excludes loops). The lattice Z d is the standard example of a Cayley graph. Notice also that the Bethe lattice T β is a Cayley graph if either S = {a 1 , . . . , a β | a 2 i = o, i = 1, . . . , β} or β = 2N and Γ = F N is a free group of N generators.
It is known that the positivity of a combinatorial isoperimetric constant α comb is closely connected with the amenability of the group Γ (this is a variant of Følner's criterion, see, e.g., [58,Proposition 12.4]).
Theorem 8.11. If G d = C(Γ, S) is the Cayley graph of a finitely generated group Γ, then α comb (Γ) = 0 if and only if Γ is an amenable group.
Notice that the class of amenable groups contains all Abelian groups, all subgroups of amenable groups, all solvable groups etc. In turn, the class of nonamenable groups includes countable discrete groups containing free subgroups of two generators. For further information on amenability and Cayley graphs we refer to [44,45,48,49,57,58].
Combining Theorem 8.11 with Corollary 4.4 and Corollary 5.2, we arrive at the following result. (i) If Γ is an amenable group, then the analysis of λ 0 (H) and λ ess 0 (H) in the case ℓ * (G) = 0 remains an open (and, in our opinion, rather complicated) problem.
(ii) The volume growth provides a number of amenability criteria. For example, groups of polynomial or subexponential growth are amenable. For further results and references we refer to [49]. (iii) Using a completely different approach, the inequality λ 0 (H) > 0 was proved recently in [13,Theorem 4.16] for Cayley graphs of free groups under the additional symmetry assumption that edges in the same edge orbit have the same length.