Self‐adjoint and Markovian extensions of infinite quantum graphs

Abstract We investigate the relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self‐adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of finite volume for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self‐adjointness of the Gaffney Laplacian — the underlying metric graph does not have finite volume ends. If, however, finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.


INTRODUCTION
This paper is concerned with developing extension theory for infinite quantum graphs. Quantum graphs are Schrödinger operators on metric graphs, that is, combinatorial graphs where edges are considered as intervals with certain lengths. Motivated by a vast amount of applications in chemistry and physics, they have become a popular subject in the last decades (we refer to [8,9,26,67] for an overview and further references). From the perspective of Dirichlet forms, quantum graphs play an important role as an intermediate setting between Laplacians on Riemannian manifolds and difference Laplacians on weighted graphs. On the one hand, being locally onedimensional, quantum graphs allow to simplify considerations of complicated geometries. On the other hand, there is a close relationship between random walks on graphs and Brownian motion on metric graphs, however, in contrast to the discrete case, the corresponding quadratic form in the metric case is a strongly local Dirichlet form and in this situation more tools are available (see [7,28,58,59] for various manifestations of this point of view). Let us also mention that metric graphs can be seen as non-Archimedian analogs of Riemann surfaces, which finds numerous applications in algebraic geometry (see [2,5,6,70] for further references).
The most studied quantum graph operator is the Kirchhoff Laplacian, which provides the analog of the Laplace-Beltrami operator in the setting of metric graphs. Its spectral properties are crucial in connection with the heat equation and the Schrödinger equation and any further analysis usually relies on the self-adjointness of the Laplacian. Whereas on finite metric graphs the Kirchhoff Laplacian is always self-adjoint, the question is more subtle for graphs with infinitely many edges. For instance, a uniform lower bound for the edge lengths guarantees self-adjointness (see [9,67]), but this commonly used condition is independent of the combinatorial graph structure and clearly excludes a number of interesting cases (the so-called fractal metric graphs). Moreover, most of the results on strongly local Dirichlet forms require completeness of a given metric space with respect to the 'intrinsic' metric (cf., for example, [74]), which coincides with the natural path (geodesic) metric in the case of metric graphs. Geodesic completeness (with respect to the natural path metric) guarantees self-adjointness of the (minimal) Kirchhoff Laplacian, however, this result is far from being optimal (see [27,Section 4] and also Section 2.4). The search for self-adjointness criteria for infinite quantum graphs is an open and -in our opinion -rather difficult problem.
If the (minimal) Kirchhoff Laplacian is not self-adjoint, the natural next step is to ask for a description of its self-adjoint extensions, which corresponds to possible descriptions of the system in quantum mechanics or, if we speak about Markovian extensions, possible descriptions of Brownian motions. Naturally, this question is tightly related to finding appropriate boundary notions for infinite graphs. Our goal in this paper is to investigate the connection between extension theory and one particular notion, namely graph ends, a concept which goes back to the work of Freudenthal [30] and Halin [38] and provides a rather refined way of compactifying graphs. However, the definition of graph ends is purely combinatorial and naturally must be modified to capture the additional metric structure of our setting. Based on the correspondence between graph ends and topological ends of metric graphs, we introduce the concept of ends of finite volume. First of all, it turns out that finite volume ends play a crucial role in describing the Sobolev spaces 1 and 1 0 on metric graphs. More specifically, we show that the presence of finite volume ends is the only reason for the strict inclusion 1 0 ⊊ 1 to hold. This in particular provides a surprisingly transparent geometric characterization of the uniqueness of Markovian extensions of the minimal Kirchhoff Laplacian as well as the self-adjointness of the so-called Gaffney Laplacian (we are not aware of its analogs either in the manifold setting or in the context of weighted graph Laplacians, cf. [35,37,45,52,61,62]). As yet the other manifestation of the fact that finite volume graph ends represent the proper boundary for Markovian extensions of the Kirchhoff Laplacian, we provide a complete description of all finite energy extensions (that is, self-adjoint extensions with domains contained in 1 , and all Markovian extensions clearly satisfy this condition), however, under the additional assumption that there are only finitely many finite volume ends. Let us stress that this class of graphs includes a wide range of interesting models (Cayley graphs of a large class of finitely generated groups, tessellating graphs, rooted antitrees, etc. have exactly one end and in this case there are no finite volume ends exactly when the total volume of the corresponding metric graph is infinite). Moreover, we emphasize that in all those cases the dimension of the space of finite energy extensions is equal to the number of finite volume ends, however, for deficiency indices, that is, the dimension of the space of self-adjoint extensions, this only gives a lower bound (for example, for Cayley graphs the dimension of the space of finite energy extensions is independent of the choice of a generating set, although deficiency indices do depend on this choice in a rather nontrivial way). On the other hand, it may happen that these dimensions coincide. The latter holds only if the maximal domain is contained in 1 , that is, if every self-adjoint extension is a finite energy extension. This is further equivalent to the validity of a certain nontrivial Sobolev-type inequality (see (1.1)). The appearance of this condition demonstrates the mixed dimensional behavior of infinite metric graphs since the analogous estimate holds true in the one-dimensional situation, but usually fails in the PDE setting.
Let us now sketch the structure of the article and describe its content and our results in greater details.
In Section 2, we collect basic notions and facts about graphs and metric graphs (Subsection 2.1); graph ends (Subsection 2.2); the minimal and maximal Kirchhoff Laplacians (Subsection 2.3); deficiency indices and their connection with the spaces of 2 harmonic and -harmonic functions (Subsection 2.4).
The core of the paper is Section 3, where we discuss the Sobolev spaces 1 () and 1 0 () and introduce the set of finite volume ends ℭ 0 () (Definition 3.8). We show that ℭ 0 () is the proper boundary for 1 functions, which can also be seen as an ideal boundary by applying * -algebra techniques (see Remark 3.14). The central result of this section is Theorem 3.12, which shows that 1 () = 1 0 () if and only if there are no finite volume ends. The latter also leads to a surprisingly transparent geometric characterization of the uniqueness of Markovian extensions of the Kirchhoff Laplacian (Corollary 5.5) as well as the self-adjointness of the Gaffney Laplacian (see Remark 5.6(ii) for details and the definition of ). Section 4 contains further applications of the above considerations. Namely, Theorem 4.1 demonstrates that deficiency indices of the minimal Kirchhoff Laplacian can be estimated from below by the number of finite volume ends. This estimate is sharp (for example, if there are infinitely many finite volume ends) and we also find necessary and sufficient conditions for the equality to hold. In particular, if there are only finitely many ends of finite volume, #ℭ 0 () < ∞, the latter is equivalent to the validity of the following Sobolev-type inequality (see Remark 4.2) ‖ ′ ‖ 2 () ⩽ (‖ ‖ 2 () + ‖ ′′ ‖ 2 () ) (1. 1) for all in the maximal domain of the Kirchhoff Laplacian. Metric graphs are locally onedimensional and the corresponding inequality is trivially satisfied in the one-dimensional case, however, globally infinite metric graphs are more complex and hence (1.1) rather resembles the multi-dimensional setting of PDEs (in particular, (1.1) does not hold true if  has a non-free finite volume end, see Proposition 4.9).
In the next sections, we focus on a particular class of self-adjoint extensions whose domains are contained in 1 (we call them finite energy extensions). These extensions have good properties and their importance stems from the fact that they contain the class of Markovian extensions (they also arise as self-adjoint restrictions of the Gaffney Laplacian). In Section 5, we show that (under some additional mild assumptions) their resolvents and heat semigroups are integral operators with continuous, bounded kernels and they belong to the trace class if  has finite total volume (Theorems 5.1 and 5.2).
In Section 6, we proceed further and show that finite volume ends is the proper boundary for this class of extensions. Namely, under the additional and rather restrictive assumption of finitely many ends with finite volume, in Subsections 6.1 and 6.2, we introduce a suitable notion of a normal derivative at graph ends (as a by-product, this also gives an explicit description of the domain of the Neumann extension, see Corollary 6.7). Section 6.3 contains a complete description of finite energy extensions and also of Markovian extensions (Theorem 6.11). Let us stress that the case of infinitely many ends is incomparably more complicated and will be the subject of future work.
In general, the inequality in (1.1) is difficult to verify/contradict and even simple examples can exhibit rather complicated behavior (see Appendix B). The only reason for which (1.1) fails to hold is the presence of 2 harmonic functions having infinite energy, that is, not belonging to 1 . Moreover, in order to compute deficiency indices of the Kirchhoff Laplacian one, roughly speaking, needs to find the dimension of the space of 2 harmonic functions and description of self-adjoint extensions requires a thorough understanding of the behavior of 2 harmonic functions at 'infinity'. Dictated by a distinguished role of harmonic functions in analysis, there is an enormous amount of literature dedicated to various classes of harmonic functions (positive, bounded, etc.), which is further related to different notions of boundaries (metric completion, Poisson and Martin boundaries, Royden and Kuramochi boundaries, etc.) and search for a suitable notion in this context (namely, 2 harmonic functions) is a highly nontrivial problem, which seems not to be very well-studied either in the context of incomplete manifolds (cf. [61,62]) or in the case of weighted graphs (see [39,45]). We further illustrate this by considering the case of rooted antitrees, a special class of infinite graphs with a particularly high degree of symmetry (see Section 7). Infinite rooted antitrees have exactly one graph end, which makes them a good toy model for our purposes. The above considerations show that the space of finite energy 2 harmonic functions is nontrivial only if a given metric antitree has finite total volume and in this case the only such functions are constants. However, adjusting lengths in a suitable way for a concrete polynomially growing antitree ( Figure 1) we can make the space of 2 harmonic functions as large as we please (even infinite dimensional!).
For a given set , # denotes its cardinality if is finite; otherwise we set # = ∞.
If it is not explicitly stated otherwise, we shall denote by ( ) a sequence ( ) ∞ =0 . ( ) is the space of bounded, continuous functions on a locally compact space . 0 ( ) is the space of continuous functions vanishing at infinity. For a finite or countable set , ( ) is the set of complex-valued functions on .  = (, ) is a discrete graph (satisfying Hypothesis 2.1).

Combinatorial and metric graphs
In what follows,  = (, ) will be an unoriented graph with countably infinite sets of vertices  and edges . For two vertices , ∈  we shall write ∼ if there is an edge , ∈  connecting with . For every ∈ , we denote the set of edges incident to the vertex by  and is called the degree (valency or combinatorial degree) of a vertex ∈ . When there is no risk of confusion about which graph is involved, we shall simplify and write deg instead of deg  . A path  of length ∈ ℤ ⩾0 ∪ {∞} is a sequence of vertices ( 0 , 1 , … , ) such that −1 ∼ holds true for all ∈ {1, … , }. The following assumption is imposed throughout the paper. Hypothesis 2.1.  is infinite, locally finite (deg( ) < ∞ for every ∈ ), connected (for any two vertices , ∈  there is a path connecting and ), and simple (there are no loops or multiple edges).
Next, let us assign each edge ∈  a finite length | | ∈ (0, ∞). We can then naturally associate with ( , | ⋅ |) = (, , | ⋅ |) a metric space : first, we identify each edge ∈  with a copy of the interval  ∶= [0, | |]. The topological space  is then obtained by 'gluing together' the ends of edges corresponding to the same vertex (in the sense of a topological quotient, see, for example, [13,Chapter 3.2.2]). The topology on  is metrizable by the natural path metric -the distance between two points , ∈  is defined as the arc length of the 'shortest path' connecting them (if or are not vertices, then we need to allow also paths which start or end in the middle of edges; the length of such paths is naturally defined by taking the corresponding portion of the interval). The metric space  arising from the above construction is called a metric graph (associated to ( , | ⋅ |) = (, , | ⋅ |)).
Note that, by definition, (, ) is a length space (see [13,Chapter 2.1] for definitions and further details). Moreover (see, for example, [40, Chapter 1.1]), a metric graph  is a Hausdorff topological space with countable base and each ∈  has a neighborhood isometric to a star-shaped set Note that deg( ) in (2.2) coincides with the combinatorial degree if belongs to the vertex set, and deg( ) = 2 for every non-vertex point of . Sometimes, we will consider  as a rooted graph with a fixed root ∈ . In this case, we denote by , ∈ ℤ ⩾0 the th combinatorial sphere with respect to the order induced by (note that 0 = { }).

Graph ends
One possible definition of a boundary for an infinite graph is the notion of the so-called graph ends (see [30; 38; 76, Section 21]). Definition 2.1. A sequence of distinct vertices ( ) ∈ℤ ⩾0 (respectively, ( ) ∈ℤ ) which satisfies ∼ +1 for all ∈ ℤ ⩾0 (respectively, for all ∈ ℤ) is called a ray (respectively, double ray). A subsequence of such a sequence is called a tail.
Two rays  1 ,  2 are called equivalent -and we write  1 ∼  2 -if there is a third ray containing infinitely many vertices of both  1 and  2 . † An equivalence class of rays is called a graph end of  and the set of graph ends will be denoted by Ω( ). Moreover, we will write  ∈ whenever  is a ray belonging to the end ∈ Ω( ).
An important feature of graph ends is their relation to topological ends of a metric graph . For locally finite graphs, there is a bijection between topological ends of a metric graph ℭ() and graph ends Ω( ) of the underlying combinatorial graph  (see [23,Section 8.6 and also pp. 277-278; 76, Section 21]; for the case of graphs which are not locally finite see [18,24]). ∈ Ω( ) such that for every sequence  representing , each contains a ray from . Moreover, the map ↦ is a bijection between ℭ() and Ω( ).
Therefore, we may identify topological ends of a metric graph  and graph ends of the underlying graph  . We will simply speak of the ends of . One obvious advantage of this identification is the fact that the definition of Ω( ) is purely combinatorial and does not depend on edge lengths. Definition 2.4. An end of a graph  is called free if there is a finite set of vertices which separates from all other ends of the graph (that is, the rays of all ends ′ ≠ end up in different connected components of  ⧵ than the rays of ). (iii) A -regular tree, ⩾ 3, has uncountably many ends, none of which is free. (iv) If  is a Cayley graph of a finitely generated infinite group , then the number of ends of  is independent of the generating set and  has either one, two, or infinitely many ends. Moreover,  has exactly two ends only if is virtually infinite cyclic (it has a finite normal subgroup such that the quotient group ∕ is isomorphic either to ℤ or ℤ 2 * ℤ 2 ). These results are due to Freudenthal [30] and Hopf [42] (see also [75]). The classification of finitely generated groups with infinitely many ends is due to Stallings [73]. Let us mention that if has infinitely many ends, then the result of Stallings implies that it contains a non-Abelian free subgroup and hence is non-amenable. For further details we refer to, for example, [32,Chapter 13]. † Equivalently,  1 ∼  2 if and only if  1 and  2 cannot be separated by a finite vertex set, that is, for every finite subset ⊂  the remaining tails of  1 and  2 in  ⧵ belong to the same connected component of  ⧵ .
(v) Let us also mention that by Halin's theorem [38] every locally finite graph  with infinitely many ends contains at least one end which is not free.
One of the main features of graph ends is that they provide a rather refined way of compactifying graphs (see [23,Section 8.6;29;76]). Namely, we introduce a topology on ∶=  ∪ ℭ() as follows. For an open subset ⊆ , denote its extensionˆto bŷ Now we can introduce a neighborhood basis of ∈ ℭ() as follows This turns into a compact topological space, called the end (or Freudenthal) compactification of .
Remark 2.6. Note that an end ∈ ℭ() is free exactly when { } is open as a subset of ℭ() (here ℭ() carries the induced topology from). This is further equivalent to the existence of a connected subgraph  with compact boundary †  such that ⊆  eventually for any sequence  = ( ) representing and ′ ∩  = ∅ eventually for all sequences  ′ = ( ′ ) representing an end ′ ≠ .
Let us mention that topological ends can be obtained in a constructive way by means of compact exhaustions. Namely, a sequence of connected subgraphs ( ) of  such that each  has finitely many vertices and edges,  ⊆  +1 for all ⩾ 0 and ⋃  =  is called a compact exhaustion of . Clearly, each  may be identified with a compact subset of . Now iteratively construct a sequence ( ) by choosing in each step a non-compact, connected component of  ⧵  satisfying ⊆ −1 . It is easy to check that each such sequence ( ) defines a topological end ∈ ℭ() and in fact all ends ∈ ℭ() are obtained by this construction. Note also that the open subsets of such representations ∼ ( ) (actually, their topological closures, since we need to add endpoints of edges which also belong to ( )) can again be identified with connected subgraphs  ( ) ∶= and we will frequently use this fact. Let us finish this section with a few more notations. Suppose  is a ray or a finite path without self-intersections in  . We may identify  with a subgraph of  and hence with a subset of , that is, we can consider it as the union of all edges of . The latter can further be identified with Also, we need to consider paths -and in particular rays -in  starting or ending at a nonvertex point. In particular, given a path ( 0 , 1 , … , ) and a point in the interior of some edge attached to 0 , ≠ 0 , 1 , we add the interval [ , 0 ] ⊆ to ( 0 , 1 , … , ). For the resulting set, we shall write ( , 0 , 1 , … , ) and call it a non-vertex path; and likewise for rays. The set of all non-vertex rays will be denoted by ℜ(). † Note that for a subgraph of  its boundary is = { ∈ ()| deg( ) < deg  ( )} and hence is compact only if # < ∞.

Kirchhoff Laplacian
Let  be a metric graph satisfying Hypothesis 2.1. Upon identifying every ∈  with a copy of the interval  = [0, | |], we denote by 2 ( ) ∶= 2 ( ; ) the 2 -space for the (unweighted) Lebesgue measure on  and introduce the Hilbert space 2 () of functions ∶  → ℂ such that The subspace of compactly supported 2 () functions will be denoted by For every ∈ , consider the maximal operator H ,max acting on functions ∈ 2 ( ) as a negative second derivative. Here and below ( ) for ⩾ 0 denotes the usual Sobolev space on (see, for example, [12,Chapter 8]). In particular, 0 ( ) = 2 ( ) and This defines the maximal operator on 2 () by  (2.9) Integrating by parts one obtains and hence 0 0 is a non-negative symmetric operator. We call its closure 0 ∶= 0 0 in 2 () the minimal Kirchhoff Laplacian. The following result is well-known (see, for example, [16,Lemma 3.9]). Lemma 2.7. Let  be a metric graph. Then * 0 = . (2.11)

Deficiency indices
In the following, we are interested in the question whether 0 is self-adjoint, or equivalently whether the equality 0 = holds true. Let us recall one sufficient condition. Define the star weight ( ) of a vertex ∈  by Theorem 2.8 [27]. If (, ) is complete as a metric space, then 0 0 is essentially self-adjoint and If a symmetric operator is not (essentially) self-adjoint, then the degree of its non-selfadjointness is determined by its deficiency indices. Recall that the deficiency subspace  ( 0 ) of 0 is defined by (2.14) The numbers n ± ( 0 ) ∶= dim  ±i ( 0 ) = dim ker( ∓ i) (2.15) are called the deficiency indices of 0 . Note that n + ( 0 ) = n − ( 0 ) since 0 is non-negative. Moreover, 0 is self-adjoint exactly when n + ( 0 ) = n − ( 0 ) = 0. , dim  ( 0 ) is constant on each connected component of the set of regular type points of 0 . Since 0 is symmetric, each ∈ ℂ ⧵ ℝ is a point of regular type for 0 . Therefore, if 0 is a point of regular type for 0 , we immediately get dim ker( ) = dim  0 ( 0 ) = n + ( 0 ) = n − ( 0 ). □ Using the Rayleigh quotient, define Noting that the operator 0 is non-negative, 0 is a point of regular type for 0 if 0 () > 0. Thus, we arrive at the following result.
The positivity of 0 () is known in the following simple situation.
It remains to note that 0 is self-adjoint exactly when n ± ( 0 ) = 0.
ture of its combinatorial spheres (see [56,Theorem 7.1]). If  is the edge graph of a tessellation of ℝ 2 , then positivity of () can be deduced from certain curvature-type quantities [65].
Finally, let us remark that ker( ) = ℍ() ∩ 2 (), where ℍ() denotes the space of harmonic functions on , that is, the set of all 'edgewise' affine functions satisfying Kirchhoff conditions (2.7) at each vertex ∈ . Note that every function ∈ ℍ() is uniquely determined by its vertex values ∶= |  = ( ( )) ∈ . Recall also the following result (see, for example, [ Remark 2.14. The above considerations indicate that in order to understand the deficiency indices of the Kirchhoff Laplacian one needs to find the dimension of the space of 2 harmonic (or, more carefully, -harmonic) functions. Moreover, in order to describe self-adjoint extensions one has to understand the behavior of 2 harmonic functions at 'infinity', that is, near a 'boundary' of a given metric graph. However, graphs admit a lot of different notions of boundary (ends, Poisson and Martin boundaries, Royden and Kuramochi boundary, etc.) and search for a suitable notion in this context (namely, 2 harmonic functions) is a highly nontrivial problem, which seems to be not very well-studied neither in the context of incomplete manifolds nor in the case of weighted graphs. Let us also mention that recently there has been a tremendous amount of work devoted to the study of harmonic functions and self-adjoint extensions of Laplacians on weighted graph (we only refer to a brief selection of articles [19,35,39,[43][44][45][46]51]).

GRAPH ENDS AND ()
This section deals with the Sobolev space 1 on metric graphs. Its importance stems, in particular, from the fact that it serves as a form domain for a large class of self-adjoint extensions of 0 .

() and boundary values
First recall that where () is the space of continuous complex-valued functions on  and Note that ( 1 (), ‖ ⋅ ‖ 1 ) is a Hilbert space when equipped with the standard norm Moreover, dom( 0 0 ) ⊂ 1 () and we define 1 0 () as the closure of dom( 0 0 ) with respect to the norm ‖ ⋅ ‖ 1 () .
Remark 3.1. If 0 0 is essentially self-adjoint, then 1 () = 1 0 (). However, the converse is not true in general. In fact this equality is tightly connected to the uniqueness of Markovian extensions of 0 and, as we shall see, it is possible to characterize it in terms of topological ends of  (see Corollary 5.5).
Note also that 1 0 () is the form domain of the Friedrichs extension of 0 0 and 0 () defined by (2.17) is the bottom of the spectrum of .
where the supremum is taken over all nonvertex paths without self-intersections.
Suppose now that  is a fixed non-vertex path without self-intersections in . Then for every ∈ , connecting and  by some finite non-vertex path  0 , we see that there exists a non-vertex path without self-intersections  such that ∈  and | | ⩾ ||∕2 (if lies on  already, then the connecting argument is superfluous and we can simply take a portion of ). Applying (3.3) to  , we easily deduce the estimate (3.2). □ Therefore, diam() ⩽ sup  || and hence  ⩽ √ coth( 1 2 diam()).
The above considerations, in particular, imply the following crucial property of 1 -functions: if  = ( ) is a ray, then exists. Indeed, upon the identification of  with the interval   = [0, ||), the latter is an immediate corollary of the description of a Sobolev space 1 in one dimension -for a bounded interval this follows from [12,Theorem 8.2] and in the unbounded case see [12,Corollary 8.9]. Moreover, this limit only depends on the equivalence class of  (indeed, for any two equivalent rays  and  ′ there exists a third ray  ′′ containing infinitely many vertices of both  and  ′ , which immediately implies that (  ) = (  ′′ ) = (  ′ )). This enables us to introduce the following notion.
It turns out that (3.5) enables us to obtain an extension by continuity of every function ∈ 1 () to the end compactification of  (see Subsection 2.2). for every sequence  = ( ) representing .
Proof. Let ∈ ℭ() and let  = ( ) be a sequence representing . Let also be the set of all non-vertex rays contained in , ⩾ 0. We proceed by case distinction. First, assume that for sufficiently large, all rays in ℜ ( ) have length at most one. If ∈ , then there exists a (non-vertex) ray  ∈ ℜ ( ) such that  = ( , 0 , … ) and its tail  ∶= ( 0 , 1 , … ) (see Definition 2.1) belong to .
By our assumption, | | ⩽ 1 and hence Since ∈ is arbitrary, this implies Since  = ( ) represents , ⋂ = ∅ and hence lim →∞ ‖ ′ ‖ 2 ( ) = 0. This implies (3.6). Assume now that for every ∈ ℤ ⩾0 there is a ray  ∈ ℜ ( ) with || > 1. Take ⩾ 0 and choose an ∈ . We can find a finite (non-vertex) path without self-intersections  ⊆ such that ∈  and | | = 1∕2 (take into account that contains at least one ray of length greater than 1). Hence, we get However, ⋂ = ∅ and hence sup ∈ | ( )| = (1) as → ∞. It remains to note that ( ) = 0. Indeed, by Theorem 2.3, for every ⩾ 0 there is a ray ∈ such that ⊆ and hence as → ∞. This finishes the proof. □ Taking into account the topology on =  ∪ ℭ(), the next result is a direct consequence of Lemmas 3.2 and 3.5. Proposition 3.6. Each ∈ 1 () has a unique continuous extension to the end compactification  of  and this extension is given by (3.5). Moreover,

Nontrivial and finite volume ends
Observe that some ends lead to trivial boundary values for 1 functions. For example, On the other hand, it might happen that all rays have finite length, however, ( ) = 0 for all ∈ 1 () (see, for example, the second step in the proof of Lemma 3.5).
We also need the following notion. Definition 3.8. A topological end ∈ ℭ() has finite volume (or, more precisely, finite volume neighborhood) if there is a sequence  = ( ) representing such that vol( ) < ∞ for some . Otherwise, has infinite volume. The set of all finite volume ends is denoted by ℭ 0 (). Here and below, vol( ) is the Lebesgue measure of a measurable set ⊆ . Remark 3.9. If ℭ() contains only one end, then this end has finite volume exactly when vol() < ∞. Analogously, if ∈ ℭ() is a free end, then there is a finite set of vertices separating from all other ends and hence this end has finite volume exactly when the corresponding connected component  has finite total volume.
If is not free, then the situation is more complicated. For example, for a rooted tree  =  the ends are in one-to-one correspondence with the rays from the root and hence one may possibly confuse the notion of a finite/infinite volume of an end with the finite/infinite length of the corresponding ray. More specifically, let be an end of  and let  = ( , 1 , 2 , … ) be the corresponding ray. For each ⩾ 1, let  be the subtree of  having its root at and containing all the 'descendant' vertices of . Then by definition has finite volume (neighborhood) if and only if there is ⩾ 1 such that the corresponding subtree  has finite total volume. In particular, this implies that  would have uncountably many finite volume ends in this case (here we assume for simplicity that all vertices are essential, that is, deg( ) > 2 for all ∈ ). In particular, | | < ∞ is a necessary but not sufficient condition for to have finite volume.
It turns out that nontrivial and finite volume ends are closely connected. Proof. It is not difficult to see that ( ) = 0 for all ∈ 1 () if has infinite volume. Indeed, assuming that there is ∈ 1 () such that ( ) ≠ 0, Lemma 3.5 would imply that there exists  = ( ) representing such that for all ∈ and some ∈ ℤ ⩾0 . However, since vol( ) = ∞, we conclude that is not in 2 () and this gives a contradiction.
Suppose now that ∈ ℭ() has finite volume. Take a sequence  = ( ) representing with vol( 0 ) < ∞. Pick a function ∈ 2 (0, 1) such that (0) = ′ (0) = ′ (1) = 0 and (1) = 1 and then define ∶  → ℂ by ∈ and both vertices of are in 0 , 0, ∈ and both vertices of are not in 0 , ( Clearly, ∈ 2 ( ) for every ∈ . Moreover, it is straightforward to check that satisfies Kirchhoff conditions (2.7) at every ∈ . By assumption, 0 is compact and hence it is contained in finitely many edges. Thus, there are only finitely many edges ∈  such that one of its vertices belongs to 0 and the other one does not belong to 0 . This implies that ∈ 2 () and, moreover, ′ ≢ 0 only on finitely many edges, which proves the inclusion ∈ dom( ) ∩ 1 (). Taking into account that ≡ 1 on for large enough , we conclude that ( ) = 1 and hence is nontrivial.
It remains to prove the second claim. Suppose that 1 , … , ∈ ℭ() are distinct nontrivial ends. Then we can find  = ( ), sequences representing , ∈ {1, … , }, such that vol( 1 0 ) < ∞ and 1 0 ∩ 0 = ∅ for all = 2, … , (see [29,Satz 3] or [24, Lemma 3.1]). Using the above procedure, we can construct a function ∈ dom( ) ∩ 1 () such that supp( ) ⊆ 0 and ( ) = 1. The latter also implies that ( 2 ) = ⋯ = ( ) = 0. □ Remark 3.11. If vol() = ∑ ∈ | | < ∞, then all ends have finite volume and the end compactifi-cation of  coincides with several other spaces, among them the metric completion of  and the Royden compactification of a related discrete graph (see [35,Corollary 4.22] and also [34, p. 1526]). Note that the natural path metric can be extended to =  ∪ ℭ() (see [34]). That is, the distance ( , ) between a point ∈  and an end ∈ ℭ() is the infimum over all lengths of rays starting at and belonging to . Similarly, the distance ( , ′ ) between two ends is the infimum over the lengths of all double rays with one tail part in and the other one in ′ . Then (, ) is a metric completion of  and is compact and homeomorphic to the end compactification of  (see [34] for further details).
The metric completion was considered in connection with quantum graphs in [16,17]; however, it can have a rather complicated structure if vol() = ∞ and a further analysis usually requires additional assumptions. Moreover, there are clear indications that metric completion is not a good candidate for these purposes.

Description of ()
Recall that the space 1 0 () is defined as the closure of dom( 0 0 ) ⊂ 1 () with respect to ‖ ⋅ ‖ 1 () . One can naturally conjecture that 1 0 () consists of those 1 -functions which vanish on ℭ(). In fact, the results of the previous two sections enable us to show that this is indeed the case. Proof. First of all, it immediately follows from Proposition 3.6 that ∈ 1 0 () vanishes at every end ∈ ℭ() (since this holds for each ∈ dom( 0 0 )). To prove the converse inclusion, we will follow the arguments of the proof of [35,Theorem 4.14]. Namely, suppose that ∈ 1 () and ( ) = 0 for all ∈ ℭ(). Without loss of generality, we may assume that is real-valued and ⩾ 0. To prove that ∈ 1 0 (), it suffices to construct a sequence of compactly supported functions Hence ∈ 1 () and for all . Let us now show that has compact support. Indeed, assuming the converse, there exist infinitely many distinct edges in  such that is non-zero on each . Taking into account (3.8), for each we can find a non-vertex point on such that ( ) > 1 . Since is compact, the sequence ( ) has an accumulation point ∈. By construction each edge ∈  contains at most one of the points . It follows that ∉  and hence ∈ is an end. On the other hand, is continuous on by Proposition 3.6 and thus ( ) ⩾ 1 , which contradicts our assumptions on .
It remains to show that converges to in 1 () as → ∞. Taking into account the above properties of , we get and hence by dominated convergence it is enough to show that → and ′ → ′ pointwise almost everywhere (a.e.) on . The first claim is clearly true since lim →∞ ( ) = for all ∈ [0, ∞). To prove the second claim, suppose that is differentiable at a non-vertex point ∈ . If ( ) > 0, then by continuity of , there is a neighborhood of such that = − 1 holds on for all sufficiently large > 0. Hence, is differentiable at with ′ ( ) = ′ ( ) for all large enough . Finally, if ( ) = 0, then for each there is a neighborhood of such that ⩽ 1 on . Hence ≡ 0 on and, in particular, is differentiable at with ′ ( ) = 0. However, since ⩾ 0 on  and is differentiable at , it follows that ′ ( ) = 0 as well. This finishes the proof. □ Combining Theorem 3.12 with Theorem 3.10, we arrive at the following fact. Remark 3.14. In the related setting of (weighted) discrete graphs, an important concept is the construction of boundaries by employing * -algebra techniques (this includes both Royden and Kuramochi boundaries, see [35,48,53,64,71] for further details and references). Finite volume graph ends can also be constructed by using this method. Indeed,  ∶= 1 () ⊂ () is a subalgebra by Lemma 3.2 and hence its ‖ ⋅ ‖ ∞ -closure ∶=  ‖⋅‖ ∞ is isomorphic to 0 (˜), where˜is the space of characters equipped with the weak * -topology with respect to. In general, finding˜for some concrete * -algebra is a rather complicated task. However, it turns out that in our situation˜coincides with ∶=  ∪ ℭ 0 (). Indeed, =  ∪ ℭ 0 () equipped with the induced topology of the end compactification is a locally compact Hausdorff space. Proposition 3.6 together with Theorem 3.10 shows that each function ∈ 1 () has a unique continuous extension to and this extension belongs to 0 (). Moreover, by Theorem 3.10, 1 () is point-separating and nowhere vanishing on and hence = 0 () by the Stone-Weierstrass theorem. Thus, the resulting boundary notion is precisely the space of finite volume graph ends. Let us also mention that is compact only if vol() < ∞ and in this case one can show that the Royden compactification of  as well as its Kuramochi compactification coincide with the end compactification (see [35;48,Theorem 7.11;49,p. 215] and also [41, p. 2] for the discrete case).

DEFICIENCY INDICES
Intuitively, deficiency indices should be linked to boundary notions for underlying combinatorial graphs. However, spectral properties of the operator 0 also depend on the edge lengths and this suggests that it is difficult to expect a purely combinatorial formula for the deficiency indices n ± ( 0 ) of 0 . Recall that throughout the paper, we always assume that  satisfies Hypothesis 2.1.

Deficiency indices and graph ends
The main result of this section provides criteria which allow to connect n ± ( 0 ) with the number of graph ends.
holds for all ∈ dom( ). It can be shown by examples that (4.3) may fail.
Before proving Theorem 4.1, let us first comment on some of its immediate consequences.  In fact, we only need to mention that by Halin's theorem [38] (see Remark 2.5(v)) and the finite total volume of , #ℭ 0 () = ∞ only if  contains a non-free end.
Recall that for a finitely generated group , the number of graph ends of a Cayley graph is independent of the generating set (see, for example, [32]). Combining this fact with the above statement, we obtain the following result. for every in the resolvent set ( ) of (see, for example, [69,Proposition 14.11]). In particular, (4.6) holds for all ∈ (−∞, 0 ()), where 0 () ⩾ 0 is defined by (2.17). Moreover, dom( ) ⊂ 1 0 () and hence the inclusion dom( ) ⊂ 1 () depends only on the inclusion ker( − ) ⊂ 1 () for some (and hence for all) ∈ ( ). Let us stress that  0 ( 0 ) = ker( ) = ℍ() ∩ 2 () and hence in the case 0 () > 0, one is interested in whether all 2 harmonic functions belong to 1 () or not, which is known to depend on the geometry of the underlying metric graph.
We also need the fact that functions in  ( 0 ) with ∈ (−∞, 0) can be considered as subharmonic functions and hence they should satisfy a maximum principle.   for all ∈ ℭ(), then ≡ 0. † A classification of groups having infinitely many ends is given in Stallings's ends theorem [73] (see also [32, Theorem 13.5.10] and Remark 2.5(iv)).
Let us stress that in the proof of Theorem 4.1 the equivalence of equality (4.2) and the inclusion dom( ) ⊂ 1 () was proved in the case when all finite volume ends are free. The next result shows that the inclusion never holds if there is a finite volume end which is not free.

Proposition 4.9. Let  be a metric graph having a finite volume end which is not free. Then there exists a function ∈ dom( ) which does not belong to 1 ().
Proof. First observe that we can restrict our considerations to the case of a metric graph  having finite total volume. Indeed, if is a non-free finite volume end of , then there exists a sequence  = ( ) representing such that vol( ) < ∞ for all . By definition, each is open and has compact boundary. Choosing  0 ⊂  as the subgraph with vertex set ( 0 ) =  ∩ 0 and edge set ( 0 ) = { ∈  | ⊂ 0 }, it is easy to see that  0 is a connected finite volume subgraph and is a non-free end of  0 (see also the notion of graph representation of an end in Section 6.1). Moreover, by construction the set  0 of boundary points (here,  0 is seen as a closed subset of ) is finite.

We conclude this section by mentioning some explicit examples.
Example 4.10 (Radially symmetric trees). Let  =  be a radially symmetric (metric) tree: that is, a rooted tree  such that for each ⩾ 0, all vertices in the combinatorial sphere have the same number of descendants ⩾ 2 and all edges between the combinatorial spheres and +1 have the same length. It is well-known that in this case is self-adjoint if and only if vol( ) = ∞ and deficiency indices are infinite, n ± ( 0 ) = ∞, otherwise (see, for example, [15,72]). Moreover, due to the symmetry assumptions, all graph ends are of finite volume simultaneously. Hence, we arrive at the equality  Remark 4.12. Both radially symmetric trees and antitrees are particular examples of the so-called family preserving metric graphs (see [11] and also [10]). Employing the results from [11] In conclusion, let us also emphasize that the example of the rope ladder graph in Appendix B shows that the assumption on horizontal edges cannot be omitted. More precisely, the rope ladder graph is a family preserving graph in the sense of [10] with exactly one graph end. However, it possesses infinitely many horizontal edges (that is, edges connecting vertices in the same combinatorial sphere) and Example B.5 shows that in general n ± ( 0 ) > #ℭ 0 (), even if the edge lengths are chosen symmetrically to the root, | + | = | − | for all ∈ ℤ ⩾0 .

PROPERTIES OF SELF-ADJOINT EXTENSIONS
The Sobolev space 1 () plays a distinctive role in the study of self-adjoint extensions of the minimal operator 0 . A self-adjoint extension˜of 0 is called a finite energy extension if its domain is contained in 1 (), that is, every function ∈ dom(˜) has finite energy, ‖ ′ ‖ 2 () < ∞. The main result of this section already indicates that finite energy self-adjoint extensions of the minimal operator (note that among those are the Friedrichs extension and, as we will see later in this section, all Markovian extensions) possess a number of important properties. Proof.
(i) Let˜be a self-adjoint lower semi-bounded extension of 0 ,˜⩾ for some ∈ ℝ. Without loss of generality, we may assume = 0. Then we can consider its positive semi-definite square root˜1 ∕2 , which is again self-adjoint and whose domain agrees with the form domain of˜. Accordingly, for all ∈ ℂ ⧵ [0, ∞) and = √ we get To prove the assertion about joint Hölder continuity, we need to take a closer look at the kernel  by adapting the proof of [3, Proposition 2.1]: as noticed before, the resolvent ( ,˜1 ∕2 ) is bounded from 2 () to ∞ () by Lemma 3.2 for any in the resolvent set of˜1 ∕2 . Applying the Kantorovich-Vulikh theorem (see, for example, [4, p. 113]) once again, we see that for all ∈  and some ( ; , ⋅) ∈ 2 () such that sup ∈ ‖ ( ; , ⋅)‖ 2 () < ∞. Moreover, observe that there exists = ( ) > 0 such that for all , ′ ∈ , where ( , ′ ) denotes the distance in the natural path metric on . Indeed, for any function ∈ 2 (), Because the semigroup generated by a self-adjoint semi-bounded extension˜is analytic, it is a bounded operator from the Hilbert space into its generator's form domain whenever Re > 0. A careful look at the proof of Theorem 5.1 shows that this is sufficient to establish that e −˜i s an integral operator; all further steps in the proof of Theorem 5.1 carry over almost verbatim to the study of semigroups. We can hence easily deduce the following result. (i) If sup  || < ∞, where the supremum is taken over all non-vertex paths without selfintersections, then the path metric has a natural extensionˆto the end compactification. Moreover, in this case (,ˆ) coincides with the metric completion of (, ). Indeed, the metric completion of (, ) is obtained by adding to  equivalence classes of rays of finite length (see [34,Section 2.3] for details) and the distance of ∈  to a boundary point is defined as the 'shortest length' of a path in the corresponding equivalence class starting at . Therefore, Theorems 5.1 and 5.2 imply that in this case the corresponding resolvent and semigroup kernels have a bounded and uniformly continuous extension to (,ˆ). However, we stress that in contrast to the case vol() < ∞ (see Remark 3.11), the topology generated bŷ on can differ from the end compactification topology if vol() = ∞. (ii) Discreteness of the spectrum of the Friedrichs extension is a standard fact in the case of finite total volume (see, for example, [16,Proposition 3.11] or [56, Corollary 3.5(iv)]). However, Theorem 5.1(ii) implies the stronger assertion that the resolvent of belongs to the trace class if vol() < ∞. Let us also stress that it is not true in general that every self-adjoint extension of will have a discrete spectrum if vol() < ∞, since in case of infinite deficiency indices such a self-adjoint extension could have a domain large enough to make compactness of the embedding of 1 () into 2 () irrelevant.
Recall that a self-adjoint extension˜of 0 is called Markovian if˜is a non-negative selfadjoint extension and the corresponding quadratic form is a Dirichlet form (for definitions and further details, we refer to [31,Chapter 1]). Hence, the associated semigroup e −˜, > 0, as well as resolvents (− ,˜), > 0, are Markovian: that is, are both positivity preserving (map nonnegative functions to non-negative functions) and ∞ -contractive (map the unit ball of ∞ (), and then by duality of () for all ∈ [1, ∞], into itself). Let us stress that the Friedrichs extension of 0 is a Markovian extension. Consider also the following quadratic form in 2 () This form is non-negative and closed, hence we can associate in 2 () a self-adjoint operator with it, let us denote it by . We will refer to it as the Neumann extension. It is straightforward to check that is a Dirichlet form and is also a Markovian extension of 0 . It turns out that Theorems 5.1 and 5.2 apply to all Markovian extensions of 0 . More specifically, the analog of the results for discrete Laplacians [39,Theorem 5.2] and Laplacians in Euclidean domains [31,Chapter 3] and Riemannian manifolds [37,Theorem 1.7] holds true for quantum graphs as well. Let us finish this section with the following observation.

FINITE ENERGY SELF-ADJOINT EXTENSIONS
It turns out that finite volume (topological) ends provide the right notion of the boundary for metric graphs to deal with finite energy and also with Markovian extensions of the minimal Kirchhoff Laplacian 0 . In particular, we are going to show that this end space is well-behaved as concerns the introduction of both traces and normal derivatives. More specifically, the goal of this section is to give a description of finite energy self-adjoint extensions of 0 in the case when the number of finite volume ends of  is finite, that is, #ℭ 0 () < ∞. Note that in this case all finite volume ends are free.

Normal derivatives at graph ends
Let = (,) be a (possibly infinite) connected subgraph of . Recall that its boundary (with respect to the natural topology on , see Subsection 2.1) is given bỹ For a function ∈ dom( ), we define its (inward) normal derivative at ∈ bỹ With this definition at hand, we end up with the following useful integration by parts formula.
Lemma 6.1. Let be a compact (not necessarily connected) subgraph of the metric graph . Then for all ∈ dom( ) and g ∈ 1 (). In particular, Proof. The claim follows immediately from integrating by parts, taking into account that satisfies (2.7). Setting g ≡ 1 in (6.3), we arrive at (6.4). □ To simplify our considerations, we need to introduce the following notion. Let ∈ ℭ() be a (topological) end of . Consider a sequence ( ) of connected subgraphs of  such that  ⊇  +1 and #  < ∞ for all . We say that the sequence ( ) is a graph representation of the end ∈ ℭ() if there is a sequence of open sets  = ( ) representing such that for each ⩾ 0 there exist and such that  ⊇ and ⊇  . It is easily seen that all graphs  are infinite (they have infinitely many edges). Moreover, graph representations ( ) of an end can be constructed with the help of compact exhaustions; in particular each graph end ∈ ℭ() has a representation by subgraphs (see Subsection 2.2). Proposition 6.2. Let  be a metric graph and let ∈ ℭ() be a free end of finite volume. Then for every function ∈ dom( ) and any sequence ( ) of subgraphs representing , the limit exists and is independent of the choice of ( ).
Proof. First of all, note that uniqueness of the limit follows from the inclusion property in the definition of the graph representations of . Hence, we only need to show that the limit in (6.5) indeed exists. Let ( ) be a graph representation of a free finite volume end ∈ ℭ 0 (). Since is free, we can assume that vol( 0 ) < ∞ and that  0 ∩ = ∅ eventually for every sequence  = ( ) representing an end ′ ≠ . First observe that =  ⧵  can again be identified with a compact subgraph of  whenever ⩽ . Indeed, if has infinitely many edges { } ⊂ , choose for each a point in the interior of the edge . Since =  ∪ ℭ() is compact, the set { } has an accumulation point ∈. By construction, ∉  and hence ∈ ⧵  = ℭ() is an end. However, we have that ∉  and recalling (2.3) and (2.4), this implies that = ′ for a topological end ′ ≠ . On the other hand, ∈  0 for all and using the properties of  0 and (2.3)-(2.4) once again, we arrive at a contradiction. Now, using (6.1) it is straightforward to verify that for all ∈ dom( ). If, moreover,  ⧵  is a connected subgraph for all ⩾ 0, then it is clear that (ii) Let us first show that (6.10) holds true for all ∈ dom( ) ∩ 1 () if g = ∈ 1 (). Take a compact exhaustion ( ) of . Then by Lemma 6.1, where  is the set of vertices of  . Note that the subgraph  itself is a connected infinite graph having finite total volume and exactly one end, which can be identified with in an obvious way. Moreover, setting  ∶=  ∩  for all ⩾ 0 and noting that  is connected for all sufficiently large , the sequence ( ) provides a compact exhaustion of  . Since for all large enough ⩾ 0, we get by applying Lemma 6.5 Hence, (6.10) holds true if g = ∈ 1 (). Now observe that a simple integration by parts implies that (6.10) is valid for all compactly supported g ∈ 1 (). By continuity and Theorem 3.12, this extends further to all g ∈ 1 0 (). Finally, settingg ∶= g − ∑ ∈ℭ 0 () g( ) for g ∈ 1 (), it is immediate to check that, by Theorem 3.12,g ∈ 1 0 (). It remains to use the linearity of Γ 0 . □ It turns out that the domain of the Neumann extension admits a simple description.
Moreover, in this case ∶= ℎ. Taking into account Proposition 6.6 and the fact that is a restriction of , we immediately arrive at (6.11). □ Our next goal is to prove surjectivity of the normal derivative map. Proposition 6.8. If  is a metric graph with #ℭ 0 () < ∞, then the mapping Γ 1 is surjective.
In fact, Proposition 6.8 will follow from the following lemma. Proof. We will proceed by contradiction. Suppose that g( ) = 0 for all g ∈ dom( ) ∩ 1 (). Then, by Corollary 6.7, dom( ) ⊆ dom( ) = dom( ) ∩ 1 (). However, both and are self-adjoint restrictions of and hence dom( ) = dom( ). Therefore, = and their quadratic forms also coincide, which implies that 1 0 () = 1 (). This contradicts Corollary 3.13 and hence completes the proof. □ Proof of Proposition 6.8. Let  , ∈ ℭ 0 () be the subgraphs of  constructed in the proof of Proposition 6.6(i). Every  is a connected graph with vol( ) < ∞ and only one end, which can be identified with . Hence we can apply Lemma 6.9 to obtain a functiong ∈ dom( ) ∩ 1 ( ) such thatg ( ) = 1. Here denotes the Kirchhoff Laplacian on  . Since #  < ∞, we can obviously extendg to a function g on  such that g ∈ dom( ) ∩ 1 () and g is identically zero on a neighborhood of each end ′ ≠ (see also the proof of Theorem 3.10). In particular, this implies that g ( ′ ) = 0 for all ′ ∈ ℭ 0 () ⧵ { }. Upon identification of with the single end of  we also have that This immediately implies surjectivity. □
Taking into account that amenable groups have finitely many ends, the above result applies to amenable finitely generated groups, which are not virtually infinite cyclic (see Remark 2.5(iv)). In a similar way one can obtain a complete description of Markovian extensions in the case of virtually infinite cyclic groups, however, they have two ends and the corresponding description looks a little bit more cumbersome and we leave it to the reader (cf. [31, p. 147]). The case of groups with infinitely many ends remains an open highly nontrivial problem.
Remark 6.13. A few remarks are in order.
(i) Let us mention that in the case when the domain of the maximal operator is contained in 1 () and  has finitely many finite volume ends (note that by Theorem 4.1 in this case n ± ( 0 ) = #ℭ 0 () < ∞), Proposition 6.11 provides a complete description of all self-adjoint extensions of 0 . Let us also mention that Proposition 6.11 provides a complete description of all self-adjoint restrictions of the Gaffney Laplacian , see Remark 5.6(ii). (ii) Some of the results of this section extend (to a certain extent) to the case of infinitely many ends. Let us stress that by Proposition 4.9 in the case when  has a finite volume end which is not free the above results would lead only to some (not all!) self-adjoint extensions of 0 . In our opinion, even in the case of radially symmetric trees having finite total volume the description of all self-adjoint extensions of 0 is a difficult problem. (iii) Similar relations between Markovian realizations of elliptic operators on domains or finite metric graphs (with general couplings at the vertices) on one hand, and Dirichlet property of the corresponding quadratic form's boundary term on the other hand, are of course wellknown in the literature (see, for example, [14, Proposition 5.1; 47, Theorem 3.5; 57, Theorem 6.1]). However, the setting of infinite metric graphs additionally requires much more advanced considerations of combinatorial and topological nature. In particular, it seems noteworthy to us that the results of the previous sections provide the right notion of the boundary for metric graphs, namely, the set of finite volume ends, to deal with finite energy and also with Markovian extensions of the minimal Kirchhoff Laplacian. In particular, this end space is well-behaved as concerns the introduction of traces and normal derivatives. (iv) Taking into account certain close relationships between quantum graphs and discrete Laplacians (see [27,Section 4]), one can easily obtain the results analogous to Theorems 4.1 and 6.11 for a particular class of discrete Laplacians on  defined by the following expression where is the star weight (2.12). Markovian extensions of weighted discrete Laplacians were considered also in [52]. On the other hand, [52] does not contain a finiteness assumption, however, the conclusion in our setting appears to be slightly stronger than in [52,Theorem 3.5], where the correspondence between Markovian extensions and Markovian forms on the boundary is in general not bijective.

DEFICIENCY INDICES OF ANTITREES
The main aim of this section is to construct for any ∈ ℤ ⩾1 ∪ {∞} a metric antitree such that the corresponding minimal Kirchhoff Laplacian 0 has deficiency indices n ± ( 0 ) = . Our motivation stems from the fact that every antitree has exactly one end and hence, according to considerations in the previous sections, 0 admits at most one-parameter family of Markovian extensions.

Antitrees
Let  = (, ) be a connected, simple combinatorial graph. Fix a root vertex ∈  and then order the graph with respect to the combinatorial spheres , ⩾ 0 (note that 0 = { }).  is called an antitree if every vertex in , ⩾ 1, is connected to all vertices in −1 and +1 and no vertices in for all | − | ≠ 1 (see Figure 1). Note that each antitree is uniquely determined by its sequence of sphere numbers ( ), ∶= # for ⩾ 0.
While antitrees first appeared in connection with random walks [25,54,77], they were actively studied from various different perspectives in the last years (see [11, 22 56] for quantum graphs and [21, Section 2] for further references).
It is clear that every (infinite) antitree has exactly one end. By Theorem 4.1, the deficiency indices of the corresponding minimal Kirchhoff Laplacian are at least 1 if vol() < ∞. On the other hand, under the additional symmetry assumption that  is radially symmetric (that is, for each ⩾ 0, all edges connecting combinatorial spheres and +1 have the same length), it is known that the deficiency indices are at most 1 (see [56,Theorem 4.1] and Example 4.11). It turns out that upon removing the symmetry assumption it is possible to construct antitrees such that the corresponding minimal Kirchhoff Laplacian has arbitrary finite or infinite deficiency indices. More precisely, the main aim of this section is to prove the following result. Theorem 7.1. Let  be the antitree with sphere numbers = + 1, ⩾ 0 ( Figure 1). Then for each ∈ ℤ ⩾1 ∪ {∞} there are edge lengths such that the corresponding minimal Kirchhoff Laplacian 0 has the deficiency indices n ± ( 0 ) = .

Harmonic functions
As it was mentioned already, every harmonic function is uniquely determined by its values at the vertices. On the other hand, ∈ () defines a function ∈ ℍ() with |  = if and only if the following conditions are satisfied: at each , 1 ⩽ ⩽ with ⩾ 0. We set −1 ∶= 0 for notational simplicity and hence the second summand in (7.1) is absent when = 0. We can put the above difference equations into the more convenient matrix form. Denote ∶= | = ( ( )) =1 for all ∈ ℤ ⩾0 and introduce matrices and ∶= diag( ) ∈ ℝ × , ∶= for all ∈ ℤ ⩾0 . Note the following useful identity where ∶= (1, … , 1) ⊤ ∈ ℝ . Hence, (7.1) can be written as follows Since is invertible, we get for all ⩾ 1. In particular, ∈ ran( −1 ( +1 * )) for all ⩾ 1, which implies that the number of linearly independent solutions to the above difference equations (and hence the number of linearly independent harmonic functions) depends on the ranks of the matrices ( +1 * ), ⩾ 1.
Let us demonstrate this by considering the following example.

7.5
Proof of Theorem 7.1 Clearly, the case of infinite deficiency indices follows from Proposition 7.7. On the other hand, since adding and/or removing finitely many edges and vertices to a graph does not change the deficiency indices of the minimal Kirchhoff Laplacian, Proposition 7.5 completes the proof of Theorem 7.1. Indeed, every antitree  can be obtained from  by first removing all the edges between combinatorial spheres 0 and and then adding + 1 edges connecting the root with the vertices in . □ Remark 7.9. Since every infinite antitree has exactly one end, Theorem 6.11(iv) implies that the Kirchhoff Laplacian 0 in Theorem 7.1 has a unique Markovian extension exactly when vol() = ∞. If vol() < ∞, then Markovian extensions of 0 form a one-parameter family explicitly given by (6.17). Note that (6.17) looks similar to the description of self-adjoint extensions of the minimal Kirchhoff Laplacian on radially symmetric antitrees obtained recently in [56].
Let us also emphasize that the antitree constructed in Proposition 7.7 has finite total volume and 0 has infinite deficiency indices, however, the set of Markovian extensions of 0 forms a one-parameter family.
Let us finish this section with one more comment. As it was proved, the dimension of the space of Markovian extensions depends only on the space of graph ends and, moreover, it is equal to the number of finite volume ends. However, deficiency indices (dimension of the space of selfadjoint extensions) are in general independent of graph ends and we can only provide a lower bound. Moreover, the above example of a polynomially growing antitree shows that the space of non-constant harmonic functions heavily depends on the choice of edge lengths (in particular, its dimension may vary between zero and infinity). In this respect, let us also emphasize that in the case of Cayley graphs of finitely generated groups the end space is independent of the choice of a generating set, however, simple examples show that the space of harmonic functions does depend on this choice.

APPENDIX A: LINEAR RELATIONS IN HILBERT SPACES
In this section, we collect basic notions and facts on linear relations in Hilbert spaces, a very convenient concept of multi-valued linear operators. For simplicity, we shall assume that  is a finite-dimensional Hilbert space, ∶= dim() < ∞.
A linear relation Θ in  is a linear subspace in  × . Linear operators become special linear relations (single valued) after identifying them with their graphs in  × . Consider linear relations in  having the form Taking into account that every linear relation in  admits one of the forms (A.1) or (A.2), this provides a description of self-adjoint linear relations in . Note also that the second condition in (A.3) is equivalent to the fact that the matrix ( | ) ∈ ℂ ×2 has the maximal rank .
By construction, deg( ) = 2 and deg( + ) = deg( − ) = 3 for all ⩾ 1. Moreover, an infinite rope ladder graph has exactly one end. Note also that a similar example was studied in [46,Section 7] (see also [33,Section 5]) in context with the construction of non-constant harmonic functions of finite energy.
We omit the proof since it is easy to check that the first condition is equivalent to the geodesic completeness of (, ) (cf. Theorem 2.8). Due to the symmetry of the underlying combinatorial graph, the gap between the above two conditions is equivalent to the fact that the corresponding lengths satisfy is based upon work from COST Action CA18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology).

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