Curvature, geometry and spectrum of planar graphs

Curvature, geometry and spectral properties of planar graphs

Matthias Keller Mathematical institute, Friedrich-Schiller-University Jena, D-07743 Jena, Germany m.keller@uni-jena.de
Abstract.

We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger’s constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies absence of finitely supported eigenfunctions for nearest neighbor operators.

1. Introduction

There is a long tradition of studying planar graphs, i.e., graphs that can be embedded in a topological surface homeomorphic to without self intersection. In particular, the curvature of tessellating graphs received a great deal of attention during the recent years, see for instance [BP1, BP2, CC, DM, Fo, H, Ke, KP, St, SY, Woe, Z] and references therein. Here, we introduce a notion of curvature for general planar graphs to study their geometry and spectral properties. In the case of tessellations, which are also called tilings, the definition coincides with the one of [BP1, BP2, H, St, Woe].

We next summarize and discuss the main results of the paper. Precise formulations can be found in the sections, where the results are proven.

The key insight of this paper is that non-positive curvature alone has already very strong implications on the structure of a planar graph, i.e.:

  • The graph is infinite and locally tessellating (Theorem 1 in Section 3).

  • The graph is locally similar to a tessellation (Theorem 2 in Section 4).

Let us comment on these points. In contrast to the statement about the infinity, positive curvature implies finiteness of the graph. This was studied in [CC, DM, H, St, SY]. Note also that locally tessellating graphs, recently introduced in [KP] (see also [Z]), allow for a unified treatment of planar tessellations and trees. The idea of (2.) is to construct an embedding of a locally tessellating graph into a tessellation that preserves crucial properties of a fixed finite subset. This embedding allows us to carry over many results known for tessellations to planar graphs. Indeed, most of the results are then direct corollaries of (1.), (2.) and the corresponding results for tessellations. In particular, we obtain for planar graphs with non-positive curvature:

  • Absence of cut locus, i.e., every distance minimizing path can be continued to infinity (Theorem 3 in Section 5.1).

  • A description of the boundary of distance balls (Theorem 4 in Section 5.2).

For regular tessellations a result similar to (3.) was obtained by Baues, Peyerimhoff in [BP1] and was later extended to general tessellations by the same authors in [BP2]. This result is considered a discrete analog of the Hadamard-Cartan theorem in differential geometry. Questions about the possible depth of the cut locus for Cayley graphs were studied under the name dead-end-depth in [CR], see also references therein. The boundary structure of distance balls, mentioned in (4.), plays a crucial role in the techniques of uk, [Z]. These concepts were later refined by [BP1, BP2] under the name admissibility of distance balls. Here, we generalize the most important statements of [BP2, Z] to planar graphs.

For negatively curved planar graphs, we prove the following:

  • Bounds for the growth of distance balls (Theorem 5 in Section 5.3).

  • Positivity and bounds for Cheeger’s constant (Theorem 6 in Section 5.4).

  • Empty interior for minimal bigons (Theorem 7 in Section 5.5).

In [BP1], a result similar to (5.) is shown for tessellations. As for (6.), positivity of Cheeger’s constant was proven by Woess [Woe] for tessellations and simultaneously by uk in a slightly more general context. Later, this was rediscovered in [H]. Here, we give new explicit bounds for Cheeger’s constant which can be interpreted in terms of the curvature. For related results, see for instance [HJL, HS, Ke, KP]. The results (6.) and (7.) can be understood as hyperbolic properties of a graph. In particular, if is the Cayley graph of a finitely generated group both results imply Gromov hyperbolicity. In general, this is not the case in our situation. However, under the additional assumption that the number of edges of finite polygons is uniformly bounded, empty interior for minimal bigons, along with Gromov hyperbolicity can be proven, see [Z] and [BP2].

Let us now turn to the spectral implications. It is well known that bounds for Cheeger’s constant imply bounds for the bottom of the spectrum of the graph Laplacian, see [BMS-T, D, DKe, Fu, M]. The Cheeger constant also plays an important role in random walks. In particular, the simple random walk is transient if Cheeger’s constant is positive, see [Ger, Woe, Woe2] and also [So]. Along with these well known relations, we show for planar graphs with non-positive curvature:

  • Triviality of the essential spectrum of the Laplacian if and only if the curvature decreases uniformly to (Theorem 8 in Section 6.1).

  • Absence of finitely supported eigenfunctions for nearest neighbor operators (Theorem 9 in Section 6.2).

Statement (8.) generalizes results of [Ke] for tessellations and of [Fu] for trees. Statement (9.) is an extension of [KLPS]. As for general planar graphs eigenfunctions of finite support can occur, this unique continuation statement shows that the analogy between Riemannian manifolds and graphs is much stronger in the case of non-positive curvature.

The paper is structured as follows. In Section 2, we give the basic definitions. Section 3 is devoted to the statement and the proof of Theorem 1. In Section 4, we construct the embedding into tessellations. Section 5 concentrates on the geometric applications (3.)-(7.) and Section 6 is devoted to spectral applications (8.) and (9.).

2. Definitions and a combinatorial Gauss-Bonnet formula

Let be a graph embedded in an oriented topological surface . An embedding is a continuous one-to-one mapping from a topological realization of into . A graph admitting such an embedding into is referred to as planar. We will identify with its image in . The faces of are defined as the closures of the connected components of . We write .

We call simple if it has no loops (i.e., no edge contains only one vertex) and no multiple edges (i.e., two vertices are not connected by more than one edge). We say that is locally finite if for every point in there exists an open neighborhood of this point that intersects with only finitely many edges. A characterization whether a Cayley graph of a group allows for a planar embedding that is locally finite was recently given in [Geo].

For the rest of this paper, we will assume that is locally finite.

We call two distinct vertices adjacent and we write if there is an edge containing both of them. We also say that two elements of , and (possibly of distinct type) are adjacent if their intersection is non-empty and we call two adjacent elements of the same type neighbors.

By the embedding, each edge corresponds uniquely to a curve (up to parametrization). We call a sequence of edges a walk if the corresponding curves can be composed to a curve. Here, we allow for two sided infinite sequences. The length of the walk is the number of edges in the sequence, whenever there are only finitely many and infinite otherwise. We refer to a subgraph, where the edges form a walk and each vertex is contained in exactly two edges as a path. The length of a path is the length of the shortest walk passing all vertices. We call vertices that are contained in two edges of a walk or a path the inner vertices of the walk or the path. A walk or a path is called closed if every vertex contained in it is an inner vertex and simply closed if all vertices are contained in exactly two edges. Note that every simply closed walk induces a path and every closed path is simply closed. We say a graph is connected if any two vertices can be joined by a path.

For the remainder of this paper, we will assume that is connected.

For a face , we call a walk a boundary walk of if it meets all vertices in and if it is closed whenever it is finite. The existence of boundary walks for all faces is ensured by the connectedness of the graph. We let the degree of a face be the length of the shortest boundary walk of , whenever admits a finite boundary walk and infinite otherwise. We define the degree of an edge as the number of vertices contained in . For a vertex , we define the degree by

The formula can be interpreted as the number of adjacent edges with degree two plus twice the number of adjacent edges with degree one. The latter are counted twice since they meet twice. A vertex with is called a terminal vertex.

To define curvature functions, we first have to introduce the corners of a planar graph . The set of corners is a subset of such that the elements satisfy . We denote for a vertex the set and for a face the set . The degree or the multiplicity of a corner is the minimal number of times the vertex is met by a boundary path of . For tessellations, the definition of corners coincides with the one in [BP1, BP2] and there every corner has degree one. However, if is not a tessellation the degree of a corner can be larger than one, (see for instance the left hand side of Figure 1 in the next section). For a vertex and a face , we have

We can think of the corners of a vertex with respect to the multiplicity as the partitions of a sufficiently small ball after removing the edges adjacent to , (where, small means that no edge is completely included in the ball and there are exactly partitions).

The curvature function on the corners is defined by

with the convention that whenever . We define the curvature function on the vertices by

By direct calculation, we arrive at

We define the curvature function on the faces by

If and if there are infinitely many vertices in with vertex degree of at least three, then the curvature of a face takes the value .

For finite subsets , , we write

Moreover, we let

We call a face a polygon if it is homeomorphic to the closure of the unit disc in and its boundary is a closed path. We call a face an infinigon if it is homeomorphic to or the upper half plane and its boundary is a path. A graph is called tessellating if the following conditions are satisfied:

  • Every edge is contained in precisely two different faces.

  • Every two faces are either disjoint or intersect precisely in a vertex or in an edge.

  • Every face is a polygon.

The additional assumption in [BP1, BP2] that each vertex has finite degree is already implied by local finiteness of . Note that the dual graph of a non-positively corner curved tessellation is also a non-positively corner curved tessellation. For a discussion, we refer to [BP1].

We call a path of at least two edges an extended edge if all inner vertices have degree two and the beginning and ending vertex, in the case that they exist, have degree greater than two. An extended edge contained in two faces with infinite degree is called regular. We introduce two weaker conditions than (T2) and (T3):

  • Every two faces are either disjoint or intersect precisely in a vertex, an edge or a regular extended edge.

  • Every face is a polygon or an infinigon.

We call a graph satisfying (T1), (T2), (T3*) strictly locally tessellating and a graph satisfying (T1), (T2*), (T3*) locally tessellating. Note that a locally tessellating graph that contains no extended edge is strictly locally tessellating. While the graphs studied in [BP1, BP2, CC, DM, H, HS, Ke, KLPS, So, St, SY, Woe] are tessellating, the results of [KP, Z] concern strictly locally tessellating graphs. We give some examples.

Examples.

(1.) Tessellations of the plane or the sphere are strictly locally tessellating. Furthermore, tessellations of embedded into via stereographic projection are also strictly locally tessellating.
(2.) Trees are strictly locally tessellating if and only if the branching number of every vertex is greater than one. In this case, the tree is negatively curved in each corner, vertex and face.
(3.) The graph with vertex set and edge set is locally tessellating, but not strictly locally tessellating and has curvature zero in every corner, vertex and face.

Next, we will prove a combinatorial Gauss-Bonnet formula. We refer to [BP1] for background and proof in the case of tessellations, see also [CC, DM] for further reference.

Let be a planar graph embedded into or . For a subset , we denote by the subgraph of induced by the vertex set , where are the edges that contain only vertices in and are the faces induced by the graph . For a finite connected subset of vertices , Euler’s formula reads

Observe that contains also an unbounded face which explains the two on the right hand side.

Proposition 1.

(Gauss-Bonnet formula) Let be planar and finite and connected. Then

Proof.

We have, by definition,

Since , the first term is equal to . As is connected, we have . Thus, the second term is equal to . Let for and . Then, by the definition of . Moreover, for each there is a unique , , such that . We conclude

By Euler’s formula, we obtain the result. ∎

Remark.

The Gauss-Bonnet formula immediately implies that a finite graph must admit some positive curvature. This is, in particular, the case for locally finite planar graphs embedded in .

We write for the length of the shortest path connecting the vertices and . For a set of vertices , we define the balls and the spheres of radius by

We define the boundary faces of by

3. Infinity and local tessellating properties of non-positively curved planar graphs

This section is dedicated to prove the first main result.

Theorem 1.

Let be a planar graph that is connected and locally finite. If one of the following conditions is satisfied

  • or

  • and is simple or

  • , each extended edge is regular and there are no terminal vertices,

then is infinite and locally tessellating. If a strict inequality holds in condition or then is even strictly locally tessellating.

The main idea of the proof can be summarized as follows. The infinity is a direct consequence of the Gauss-Bonnet formula. Moreover, assuming a graph is not locally tessellating, we will construct a finite planar graph with smaller or equal curvature which can be embedded into . By the Gauss-Bonnet formula, this graph must admit some positive curvature. As the curvature is smaller or equal, the original graph must have some positive curvature as well.

We start with analyzing the pathologies that occur in general planar graphs that are not locally tessellating. Let be a planar graph that is connected and locally finite. A face is called degenerate if it contains a vertex such that . Note that this is in particular the case if is contained in three or more boundary edges or includes and an edge that is included in no other face than . A pair of faces is called degenerate if consists of at least two connected components. Figure 1 shows examples of degenerate faces. We denote by the set of all faces that are degenerate or are contained in a degenerate pair of faces.

Figure 1. Examples of a degenerate face and a degenerate pair of faces .
Lemma 1.

Let be a simple, planar graph that is connected, locally finite and contains no terminal vertices. If , then there is a finite and connected subgraph which is bounded by a simply closed path and satisfies the following property: there are at most two vertices such that each path connecting and meets either or . In particular, , lie in the boundary path of .

Proof.

The proof consists of two steps. Firstly, we find a finite set that contains at most two vertices with the asserted property. We have to deal with the case of a degenerate face and a degenerate pair separately. Secondly, we find a subgraph of that has a simply closed boundary.

Figure 2. The figure shows how to isolate a finite subset , whenever there is degenerate pair of faces or a degenerate face.

We start with the case of a degenerate pair of faces . Let be two vertices that are contained in different connected components of the intersection of and . Let be a simple curve that lies in and connects and , where . Similarly, let be a simple curve in connecting and . Composing these curves, we obtain a simply closed curve which divides the plane by Jordan’s curve theorem into a bounded and an unbounded component. We denote the bounded component by . The set can be connected to only by walks which meet or . For an illustration see the left hand side of Figure 2.
We now turn to the case when there is no degenerate pair of faces. As , there must be a degenerate face . Let be a vertex in with . We pick an open simply connected neighborhood of that contains no other vertex except for and is connected for all edges adjacent to . Let and be two arbitrary points in two different connected components of . We connect and by a simple curve in and connect and also by simple curves that lie in the corresponding connected component of . By composition, we obtain a simply closed curved and by Jordan’s curve theorem we get a bounded set . Note that can be connected to only by walks which meet . For an illustration see the right hand side of Figure 2.
In both cases, we identified a set which is finite and connected since the graph is locally finite and connected. Now, we find a subset such that is bounded by a simply closed path. By construction, lies in an unbounded face of , which we denote by and which has a finite boundary walk. We start from walking around . The walk might visit certain vertices several times. The vertex is visited at least twice, since the walk is finite. We pick a subsequence of the walk such that no vertex is met twice except for the starting and ending vertex, which we denote by . In Figure 3, this is illustrated (with ).

Figure 3. The right hand side shows an enumeration of the boundary walk around illustrated on the left hand side. To isolate the subgraph , we pick the simply closed subwalk from to .

The sequence of edges forms a closed path and encloses a subgraph, which we denote by . Note that contains which might be equal to and also might contain which we denote in this case by . Nevertheless, and are the only vertices in that can be connected to by edges of . Thus, has the desired properties and we finished the proof. ∎

The next lemma is the main tool for the proof of Theorem 1. It shows that the existence of degenerate faces or pairs implies positive curvature.

Lemma 2.

(Copy and paste lemma) Let be a simple, planar graph that is connected, locally finite, contains no terminal vertices and all extended edges are regular. If , then there are and such that and .

Proof.

The idea of the proof is easy to illustrate. We assume and take the subgraph of Lemma 1. We make copies of , paste them along the boundary and embed the resulting graph into . See Figure 4 below. We then show that the curvature of this graph compared to the curvature in does not increase, as long as we make enough copies of . The statement is then implied by the Gauss-Bonnet formula.

Figure 4. An illustration of the copy and paste procedure.

Assume . By Lemma 1, there is a finite subgraph of which is enclosed by a closed path . The vertex degree in differs from in at most two vertices, which we denote by and . (If there is only one vertex, then we choose another vertex arbitrarily in the boundary path of .) Let be the edges of the boundary path of starting and ending at , i.e., . Moreover, let such that . Denote and . We take two copies and of . We paste them along the edges of the subpaths and (where corresponds to in for ), i.e., we identify the edges in with the edges in . We denote the resulting graph by . Note that the edges of the boundary path of are the set . Denote and . Now, let be an integer, that will be quantified later. We take copies of and paste them along the subpaths and for . We embed the resulting planar graph into and we finally paste and . We denote the resulting graph embedded in by . In Figure 4, the procedure is illustrated.
We will make the following observations which are implied by the construction:

  • We can identify each corner in uniquely with a corner in and and they have the same multiplicity. For each such corner in or there are exactly corresponding corners in .

  • We can identify each face in uniquely with a face in and . Each bounded face in can be identified uniquely with a face in and exactly corresponding faces in . Moreover, the face degree of corresponding faces is the same in , and

We now quantify which was introduced above. Let

where denotes the smallest integer that is greater than for and denotes the vertex degree in . There is one more observation:

  • We can identify each vertex in uniquely with a vertex in and in . Each of these vertices in and either corresponds to , or to exactly vertices of the former boundary path of or to exactly vertices in . Moreover, the vertex degree of a vertex in is at least the vertex degree of the corresponding vertex in and it is at least three.

The statement about the corresponding vertices in (O3) follows by construction. To check the statement about the vertex degrees in (O3), one has to consider three cases. The statement is clear for vertices in that are not contained in the boundary path of . For the vertices , the statement follows by the choice of . For any other vertex , the vertex degree in is equal to , which yields the statement. Finally, we have for all since we assumed that all extended edges in are regular and all faces contained in are bounded.
In the following, we will not distinguish between the objects we identified in (O1), (O2), (O3). An obvious consequence of (O1), (O2) and (O3) is that for a corner and the corresponding corner we have

We now find that satisfies . Denote the unbounded face in by and set which we can identify with a subset of and identical subsets of by (O2). Thus, we conclude by the Gauss-Bonnet formula and the inequality above

Therefore, .

To find with , we assume that all vertices that correspond to a vertex in the boundary path of satisfy . Otherwise, we have for . We first show for . To do so, we check three cases: Firstly, the statement is clear for vertices that are not contained in the boundary path of . Secondly, for that is contained in the boundary path of , we have . Hence,

where is the unique face adjacent to in for which there is no corresponding face in . Note that corresponds to in and . The second equality follows since for any other corner in there are exactly two corresponding corners with the same multiplicity in . The last inequality is due to the assumption for vertices contained in . Thirdly, we have to check for . In the following inequality, we estimate , then use from the definition of , to obtain

As , we have for . Thus, we have shown for all

We finish the proof by identifying a subset that satisfies and, by the identification (O3), . Let be the vertices of that are not in the vertex set of the boundary path of . The Gauss-Bonnet formula and (O3) yield, since we assumed for

Therefore, . If , then we are done. Otherwise, we arrived at a contradiction to the assumption for . Hence, we conclude for for the vertex that gives the contradiction. ∎

The next lemma shows that the absence of degenerate faces and degenerate pairs characterizes whether a ”nice“ graph is locally tessellating.

Lemma 3.

Let be a simple, planar graph that is connected, locally finite, contains no terminal vertices and each extended edge is regular. Then, is locally tessellating if and only if . In this case, each corner has multiplicity one and

Proof.

Obviously, if is non-empty, then at least one of the conditions (T1), (T2*), (T3*) on page 2 is violated. On the other hand, the absence of degenerate faces implies that each corner has multiplicity one. This has the following consequences: Firstly, no edge can be included in only one face which is (T1). Secondly, the boundary of each face is a path (i.e., there is a boundary walk of the face which is meeting every vertex only once) and, hence, (T3*) follows. Thirdly, the formula for the curvature now follows from the definition. Finally, the absence of degenerate pairs of faces implies (T2*) as extended edges are assumed to be regular. ∎

Next, we show that non-positive curvature implies that the graph is ”nice“. The proof follows from straightforward calculation.

Lemma 4.

Let be a planar graph that is connected and locally finite.

  • If , then is simple, admits no terminal vertices and each extended edge is regular. If , then there are no extended edges.

  • If , then admits no terminal vertices and each extended edge is regular. If , then there are no extended edges.

  • If , then is simple.

We come now to the proof of Theorem 1.

Proof of Theorem 1..

If is finite, then by the Gauss-Bonnet formula, it must admit some positive curvature.
Lemma 4 implies that the assumptions of Lemma 2 are satisfied. Hence, by Lemma 2, we have and, by Lemma 3, we obtain that is locally tessellating. Moreover, by Lemma 4, in the case of or the graph admits no extended edges and, thus, is strictly locally tessellating. ∎

4. Embedding of a locally tessellating graph into a tessellation

In this section, we construct an embedding of a locally tessellating graph into a tessellating supergraph, which leaves crucial properties of a subset of vertices invariant. This embedding allows us to carry over many results for tessellations to planar graphs in the forthcoming sections. We start the section with an extension of a proposition of Higuchi [H], to planar graphs. Then, after presenting the construction, we will extract some important properties of the tessellating supergraph.

Proposition 2.

Let be a simple, planar graph that is connected and locally finite. If for all , then . The maximum is achieved for vertices with degree that have adjacent faces with degrees exactly , and .

Proof.

By Theorem 1 every non-negatively curved graph is locally tessellating. In [H, Proposition 2.1] it is shown that on implies on for tessellating graphs and that the maximum is achieved as it is claimed above. The proof consists of a list of all relevant cases. The list is ordered by the vertex degree and a vector where are the degrees of the faces adjacent to the vertex. Since we allow for unbounded faces, we have to check some additional cases: For , the curvature for with is smaller or equal to . For , the curvature for is zero and for with and the curvature is smaller or equal to . For , the curvature of and is zero, while the curvature of with , is smaller or equal to and with , it is smaller or equal to . ∎

We now come to the construction of the embedding. Let be a simple, locally tessellating graph that satisfies . Let be a finite set of vertices simply connected, i.e., both subgraphs and are connected. The construction consists of two steps. In the first step, we add binary trees to certain vertices. In the second step, we close the unbounded faces by adding ”horizontal“ edges.

Step 1: To any vertex with that is adjacent to infinigons, we attach binary trees. To do so, we embed every one of these tree into a different infinigon and then connect the roots of the trees and by edges. With slight abuse of notation, we denote the face set of the resulting graph also by .

Step 2: We choose the closing parameter, that is the size at which unbounded faces are closed by a ”horizontal“ edge. Let . For define

By induction over , we perform the following procedure: For every unbounded face , we connect the two vertices in by an edge whenever . (Note that the uniqueness of the two vertices follows since extended edges are regular.) We denote, with slight abuse of notation, the face set of the modified graph after each induction step again by .

This yields a limiting graph that we denote by . Obviously, is a super-graph of , i.e., , . Therefore, it is natural to talk about corresponding vertices in and . In particular, we will denote for a vertex the corresponding vertex in by and for a subset we denote the corresponding subset in by .

Theorem 2.

Let be a simple, connected, locally tessellating graph that satisfies , be finite and simply connected and . Then, the graph constructed above is a tessellation and satisfies the following assertions:

  • If is not adjacent to an infinigon or , then . Otherwise, , where is the number of adjacent infinigons. Moreover, if for all , then edges are added to if and only if is the inner vertex of an extended edge.

  • The embedding of into the supergraph is a graph isomorphism of the subgraphs and , (i.e., the adjacency relations of corresponding vertices and remain unchanged). If on , then the embedding is even a graph isomorphism of the subgraphs and .

  • The distance of two vertices in equals the distance of the corresponding vertices in .

  • If , then .

  • If , then whenever .

Proof.

It is obvious from the construction that is a tessellating graph.

(G1): Edges are added to vertices in only in Step 1. This is exactly the case, if the vertex is adjacent to at least one infinigon and . Then, as many edges are added as there are adjacent infinigons. If for all and then for all . If is adjacent to an infinigon, then and both adjacent faces are infinigons.

(G2): The first statement follows since we do not connect or disconnect vertices within . The second one follows from the first one and (G1).

(G3): Note that, in Step 2, we add edges and create paths in that are not in . By definition of , such a path in connecting vertices in has at least the length . Therefore, the distance of in is at least the distance of in . Moreover, by (G2) the distance does not increase either.

(G4): We consider three types of corners: Firstly, consider that is the corresponding corner of some with . Clearly, by (G1). Secondly, let be such that is the corresponding vertex of some vertex and is created in Step 2 by closing an infinigon. Obviously, and by construction and the definition of . Therefore, Thirdly, let be a corner of a vertex that was added with a binary tree in Step 1. In this case, . Obviously, by construction and by definition . Therefore,

(G5): We consider three cases: Firstly, let be the corresponding vertex of some vertex to which binary trees were added in Step 1. Then, by (G1). This gives a total of new faces each of which has face degree after being closed in Step 2. Therefore,

Secondly, let be the corresponding vertex of some to which no edges were added in Step 1. If is not adjacent to an infinigon, then no edges are added in Step 2 either. Thus, . Otherwise, (as, otherwise, edges were added in Step 1). Due to planarity, at most two edges were added to in Step 2, i.e., . Moreover, for a face adjacent to , there is either a corresponding face and or was created in Step 2 from an infinigon, in which case by definition of . By these considerations, we get

where the last inequality follows by the definition of . We have as by Proposition 2 whenever . Thirdly, let be a vertex that has no corresponding vertex in , i.e., it is a vertex of a binary tree which was added in Step 1. Note that, in this case, and by Step 1 and Step 2. Moreover, by definition of , all faces adjacent to satisfy . We get . ∎

5. Geometric Applications

In this section, we discuss some applications of the fact that every non-positively curved planar graph is locally tessellating. Indeed, by the embedding constructed in the previous section, most of statements for locally tessellating graphs are now direct consequences of the results for tessellations. These results concern the absence of cut locus, the boundary structure of distance balls, estimates for the growth of distance balls, bounds and positivity of Cheeger’s constant and empty interior of minimal bigons.

5.1. Absence of cut locus

The cut locus of a vertex of a graph is the set of all vertices, where the distance function attains a local maxima. In contrary, empty cut locus for all vertices implies that geodesics can be continued ad infinitum.

For non-positively curved tessellation, a corresponding result can be found in [BP2]. Note that in [BP1, BP2] the results and proofs are given for the metric space considering the distance function on the faces of the graph. However, the results are true for the metric space of vertices as well, since the dual graph of a non-positively corner curved tessellation is again a non-positively corner curved tessellation.

Theorem 3.

Let be a planar graph that is connected, locally finite and satisfies . Then, the metric space has no cut locus.

Proof.

By Theorem 1, the graph is locally tessellating. Suppose there is with non-empty cut locus and suppose is in the cut locus of . Then, by the definition of the cut locus, all adjacent vertices of have smaller or equal distance to . Consider a simply connected set of vertices that contains all vertices of the paths of minimal length from to and all adjacent vertices of . By (G2), (G3) of Theorem 2, there is a tessellation such that the distances of corresponding vertices in and agree. Let be the corresponding vertices to in . Then, is in the cut locus of in . By (G4), we have . This leads to a contradiction to [BP1, Theorem 1] which guarantees absence of cut locus for tessellations under the assumption of non-positive corner curvature. ∎

5.2. The boundary of distance balls

Non-positive corner curvature has very strong implications on the boundary structure of distance balls. In particular, the concept of admissibility introduced in [BP1, BP2] captures important aspects of the boundary behavior. Since this concept is quite involved, we only derive some of its most important consequences.

In [Z], some of these statements were already proven under various assumptions which all imply . In particular, these statements are used there to prove positivity of Cheeger’s constant. Here, we will use these properties to prove absence of finitely supported eigenfunctions of nearest neighbor operators on planar non-positively curved graphs.

Theorem 4.

Let be a planar graph that is locally finite and satisfies . Let , and denote and .

  • Every vertex in is adjacent to at least one vertex in .

  • Every vertex in is adjacent to at most two vertices in