Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, Wojciechowski.


Introduction
Open bounded sets in Euclidean space provide an important and well studied instance of spectral geometry. Recently, discrete analogues of such sets have become a focus of attention [6,10,11]. In fact, [6] proposes the concept of canonically compactifiable graphs as graphs with strong intrinsic compactness properties. By definition, a graph is called canonically compactifiable if all functions of finite energy are bounded. For such graphs, there is a natural compactification, namely, the Royden compactification. Clearly, in order to study the geometry of such graphs, it is desirable to understand metric compactness features of such graphs. Note that every canonically compactifiable graph is also uniformly transient in the sense of [10]. As shown in [6], total boundedness with respect to common metrics such as the resistance metric or a standard length metric implies that the underlying graph is canonically compactifiable but the converse is not true. So, [6] leaves open the question of a metric compactness characterization of such graphs. Still, a candidate for such a characterization is proposed there. More specifically, it is shown that total boundedness with respect to all finite measure intrinsic metrics implies that the graph is canonically compactifiable and the converse is shown to hold for locally finite trees. The converse for general graphs, however, remained open and is posed as a problem in [6]. The first main result of this note (Theorem 1) solves this problem. In particular, by combining the main result with the mentioned result of [6] we obtain that a graph is canonically compactifiable if and only if it is totally bounded with respect to all finite measure intrinsic metrics (Corollary 1).
To put this result in perspective, we briefly discuss the relevance of intrinsic metrics next. Intrinsic metrics for strongly local Dirichlet forms were introduced in [13] and have subsequently played a fundamental role as they allow for a study of intrinsic spectral geometry of such forms. For general regular Dirichlet forms a concept was only proposed recently in [5] (see [3,4,8,14] for independent related studies on graphs). A special feature of the case of general Dirichlet forms is well worth pointing out: While in the strongly local case there is a maximal intrinsic metric there are, in general, several incomparable intrinsic metrics on graphs [5]. Hence, in general, one cannot expect that it will be sufficient to consider only one intrinsic metric for graphs. Recent years have witnessed rather successful applications of intrinsic metrics in order to understand spectral geometry of graphs, see the mentioned works as well as e.g. [1,2,7]. Given this, it is very natural to look for a characterization of the intrinsic compactness property of canonical compactifiability in terms of intrinsic metrics. Theorem 1 provides such a characterization. In the last section we provide an answer to another question raised in [6]. This question concerns an algebraic characterization of canonically compactifiable graphs. More specifically, in [6] it is shown that the set of functions of finite energy is an algebra if the graph is canonically compactifiable and we show that the converse is also true (Theorem 2).

Background
In this section we first introduce the necessary notations and recall basic facts shown in [6] (see [9] for a description of the general setting as well).
A weighted graph G = (X, b) consists of a nonempty countable set X of nodes and a symmetric edge weight function b : X × X → [0, ∞) that vanishes on the diagonal and satisfies the summability condition Similarly, a graph is called connected, if all of its nodes are connected. This graph structure gives rise to a quadratic form that assigns to any function f : X → C its Dirichlet energỹ and consequently defines the functions of finite energỹ This set is closed with respect to addition, sincẽ The graph (X, b) is called canonically compactifiable, if all functions of finite energy are bounded, i.e.,D ⊆ l ∞ (X).
In the rest of this note we will only look at connected graphs. We can do this since a graph is canonically compactifiable if and only if it has finitely many connected components (i.e. equivalence classes with respect to connectedness) and every connected component is canonically compactifiable. Pseudo metrics are symmetric functions σ : X × X → [0, ∞] that vanish on the diagonal and satisfy the triangle inequality σ(x, z) ≤ σ(x, y) + σ(y, z). Any pseudo metric σ naturally induces a distance from any nonempty subset and the diameter of U ⊆ X by diam σ (U ) := sup x,y∈U σ(x, y).
Whenever (X, b) is a graph and m is a measure on X (i.e. an additive function P(X) → [0, ∞] induced by a node weight function X → (0, ∞)), a pseudo metric σ is called intrinsic with respect to the measure m, if the inequality holds for all x ∈ X. Any graph (X, b) comes with a metric ̺ defined as Given this definition, it is easy to see that any function f of bounded Dirichlet energy satisfies |f (x) − f (y)| ≤Q(f ) 1/2 ̺(x, y).
Indeed, the inequality is optimal; the definition of ̺ gives that it is characterized by for any C > 0. The metric ̺ is tied to canonical compactifiability, as Theorem 4.3 in [6] proves that a graph (X, b) is canonically compactifiable if and only if it is bounded with respect to ̺, i.e. diam ̺ (X) < ∞. This will be used below.
Remark. This metric is closely tied to the resistance metric r by ̺ 2 = r (it is shown that r is a metric for locally finite graphs in [6], Theorem 3.19 and for general graphs in [12]). The metric is also related to intrinsic metrics (see Theorem 3.13 in [6]).

Characterization via intrinsic metrics
In this section, we provide a characterization of canonical compactifiability via intrinsic metrics. A key step is the subsequent lemma. It provides an estimate for the energy of the distance to a set with respect to an intrinsic metric, which may be of interest in other contexts as well. A weaker bound (by m(X) instead of 2m(X \ U )) is given in [6], Propositions 3.10 and 3.11. Lemma 1. Let G = (X, b) be a graph and let σ be an intrinsic metric with respect to a finite measure m on X. Choose a nonempty subset U ⊂ X. Theñ Proof. Immediately, we deduce σ U (x) = 0 for all x ∈ U and |σ U (x) − σ U (y)| ≤ σ(x, y) for all x, y ∈ X. These properties already imply the desired bound on the Dirichlet energy of σ U : Here, we use that σ is an intrinsic metric in the last estimate.
Theorem 1. Let (X, b) be a canonically compactifiable graph and consider a pseudo metric σ which is intrinsic with respect to a finite measure m. Then (X, b) is totally bounded with respect to σ.
Proof. Fix an arbitrary ε > 0. We have to find a finite subset S of X with σ(x, S) < ε for all x ∈ X. For δ > 0, define the set U δ = {x ∈ X : m({x}) < δ}. As m is finite and the graph is canonically compactifiable (i.e. diam ̺ (X) < ∞ holds by the discussion above), we can choose δ > 0 such that We now claim that the set S := X \ U δ has the desired properties: Indeed, the set is finite as we clearly have |S| ≤ m(X) δ < ∞. Moreover, σ(x, S) < ε holds as can be seen as follows: Consider the function Lemma 1 helps us estimate the Dirichlet energy of this function: Now, recall the inequality |f (x) − f (y)| ≤Q(f ) 1/2 ̺(x, y) and pick an arbitrary point o ∈ S to see This finishes the proof.
Combining the previous theorem with [6], Corollary 4.5 we obtain the following characterization of canonically compactifiable graphs. Corollary 1. A graph (X, b) is canonically compactifiable if and only if X is totally bounded with respect to any metric σ that is intrinsic with respect to a finite measure.

Algebraic characterization
In this section, we will prove that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra (with the usual pointwise addition and multiplication of functions). Since Lemma 4.8 in [6] already states that the space of functions of finite Dirichlet energy is an algebra if the underlying graph is canonically compactifiable, we will focus on the other direction.
Theorem 2. Let G = (X, b) be a graph. If the space of functions of finite Dirichlet energyD is an algebra, the graph G is canonically compactifiable.
Proof. We analyze graphs that are not canonically compactifiable and find functions f ∈D such thatQ(f 2 ) = ∞, implying thatD cannot be an algebra. Let G be a graph that is not canonically compactifiable and fix an arbitrary node o ∈ X. We know that ̺ is unbounded on G, since G is not canonically compactifiable. Select an infinite sequence of nodes (x n : n ∈ I ⊆ N) such that 8 n < ̺(x n , o) ≤ 8 n+1 (if there is no such x n for a certain n, just omit this index). Now, find functions f n ∈D that satisfy f n (o) = 0, f n (x n ) = 4 n , 0 ≤ f n ≤ 4 n andQ(f n ) ≤ 4 −n .
These exist, because where the additional condition 0 ≤ f ≤ 4 n can be introduced sinceQ is a Dirichlet form.
Combining Theorem 2 with Lemma 4.8 in [6] then yields the following characterization. Remark. Let us emphasize that our results do not assume local finiteness of the graph.