Geometric and spectral properties of locally tessellating planar graphs
aa r X i v : . [ m a t h . SP ] M a y Geometric and spectral properties of locallytessellating planar graphs
Matthias Keller ∗ TU ChemnitzFakult¨at f¨ur MathematikD-09107 Chemnitz, Germany Norbert Peyerimhoff † Department of Math. SciencesUniversity of DurhamDurham DH1 2LE, UKNovember 20, 2018
Abstract
In this article, we derive bounds for values of the global geometry of locallytessellating planar graphs, namely, the Cheeger constant and exponentialgrowth, in terms of combinatorial curvatures. We also discuss spectralimplications for the Laplacians.
A locally tessellating planar graph G is a tiling of the plane with all faces tobe polygons with finitely or infinitely many boundary edges (see Subsection2.1 for precise definitions). The edges of G are continuous rectifiable curveswithout self-intersections. Faces with infinitely many boundary edges are calledinfinigons and occur, e.g., in the case of planar trees. The sets of vertices, edgesand faces of G are denoted by V , E and F . d ( v, w ) denotes the combinatorialdistance between two vertices v, w ∈ V , where each edge is assumed to havecombinatorial length one.Useful local concepts of the graph G are combinatorial curvature notions.The finest curvature notion is defined on the corners of G . A corner is a pair( v, f ) ∈ V ×F , where v is a vertex of the face f . The set of all corners is denotedby C . The corner curvature κ C is then defined as κ C ( v, f ) = 1 | v | + 1 | f | − , where | v | and | f | denote the degree of the vertex v and the face f . If f is aninfinigon, we set | f | = ∞ and 1 / | f | = 0. The curvature at a vertex v ∈ V isgiven by the sum κ ( v ) = X ( v,f ) ∈C κ C ( v, f ) = 1 − | v | X f : v ∈ f | f | . ∗ e-mail: [email protected] † e-mail: norbert.peyerimhoff@durham.ac.uk W ⊂ V we define κ ( W ) = P v ∈ W κ ( v ). These combinatorial cur-vature definitions arise naturally from considerations of the Euler characteristicand tessellations of closed surfaces, and they allow to prove a combinatorialGauß-Bonnet formula (see [BP1, Thm 1.4]). Similar combinatorial curvaturenotions have been introduced by many other authors, e.g., [Gro, Hi, St, Woe].The aim of this paper is to establish connections between local curvatureconditions and characteristic values of the global geometry of the graph G , inparticular the exponential growth and Cheeger constants. For a finite subset W ⊂ V , let vol( W ) = P v ∈ W | v | . The exponential growth is defined as follows(note that the value µ ( G ) does not depend on the choice of center v ∈ V ): Definition 1.
The exponential growth µ ( G ) is given by µ ( G ) = lim sup n →∞ log vol( B n ( v )) n , where B n ( v ) = { w ∈ V | d ( v, w ) ≤ n } denotes the (combinatorial) ball of radius n about v . We also consider the following two types of Cheeger constants.
Definition 2.
Let α ( G ) = inf W ⊆ V , | W | < ∞ | ∂ E W || W | and e α ( G ) = inf W ⊆ V , | W | < ∞ | ∂ E W | vol( W ) where ∂ E W is the set of all edges e ∈ E connecting a vertex in W with a vertex in V\ W . α ( G ) is called the physical Cheeger constant and e α ( G ) the combinatorialCheeger constant of the graph G . The attributes physical and combinatorial in the previous definition are mo-tivated by the fact that these Cheeger constants are closely linked to two typesof Laplacians: The physical Laplacian is used frequently in the community ofMathematical Physicists and is defined as follows:(∆ ϕ )( v ) = | v | ϕ ( v ) − X w ∼ v ϕ ( w ) . (1)Note that ∆ is an unbounded operator if there is no bound on the vertex degreeof G . The combinatorial Laplacian e ∆ is a bounded operator and appears in thecontext of spectral geometry (see, e.g., [DKa, DKe, Woe2]):( e ∆ ϕ )( v ) = ϕ ( v ) − | v | X u ∼ v ϕ ( u ) . (2)Both operators are defined in and are self-adjoint with respect to different l -spaces (see Subsection 2.2). In the case of fixed vertex degree, both operatorsare multiples of each other.Our main geometric results are given in Subsection 2.3, where we • provide lower bounds for both Cheeger constants in terms of combinatorialcurvatures (see Theorem 1 below),2 provide upper bounds for the exponential growth in terms of an uppervertex bound (see Theorem 2 below).Even though Theorem 2(b) is formulated in terms of bounds on vertex andface degrees, it can also be considered as an estimate in terms of combinatorialcurvature, as is explained in the remark following the theorem. In fact, theproof is based on the corresponding curvature version.Now we discuss connections to the spectrum. The Cheeger constant and theexponential growth were first introduced in the context of Riemannian manifoldsand were useful invariants to estimate the bottom of the (essential) spectrumof the Laplacian (see [Che] and [Br]). An analogous inequality between theCheeger constant and the bottom of the spectrum in the discrete case of graphswas first proved by [Do] and [Al]. This inequality is also useful in the studyof expander graphs. [Al] noted also the connection between this inequalityand the Max Flow-Min Cut Theorem (see also [Chu] and [Gri]). For otherconnections between isoperimetric inequalities and lower bounds of eigenvaluesin both continuous and discrete settings see, e.g., [CGY].The best results about the relations between the combinatorial Cheeger con-stant, the exponential growth, and the bottom e λ ( G ) and e λ ess ( G ) of the (essen-tial) spectrum of the combinatorial Laplacians e ∆ are due to K. Fujiwara (see[Fu1] and [Fu2]):1 − p − e α ( G ) ≤ e λ ( G ) ≤ e λ ess ( G ) ≤ − e µ ( G ) / e µ ( G ) . (3)These estimates are sharp in the case of regular trees. Using these estimatesand Theorems 1 and 2, we obtain • lower and upper estimates on the bottom of the (essential) spectrum of thecombinatorial Laplacian in terms of combinatorial curvature (see Corol-laries 1 and 2).Since there are estimates to compare the bottom of the (essential) spectrum ofthe combinatorial Laplacian with the physical Laplacian (see for instance [Ke])these results can be also formulated for the physical Laplacian.A lower estimate for the bottom of the essential spectrum of the combinato-rial Laplacian via the combinatorial Cheeger constant at infinity can be foundin [Fu2, Cor. 3]. This yields a discrete analogue for the combinatorial Laplacianof the result in [DL] about the emptiness of the essential spectrum for completesimply connected manifolds with curvature converging to minus infinity. Corre-sponding results about the emptiness of the essential spectrum for the physicalLaplacian can be found in [Ke, Woj].Finally, let us discuss two other interesting types of eigenfunctions, namely, strictly positive eigenfunctions and finitely supported eigenfunctions , and illus-trate all concepts in two examples.For the discrete case of a graph, it was shown in [DKa, Prop. 1.5] thatthe equation e ∆ f = λf has a positive solution if and only if λ ≤ e λ ( G ). Thischaracterisation of the bottom of the spectrum was well known before in thecontext of Riemannian manifolds (see, e.g., [Sull] and the references therein).In the reverse direction, this characterisation might be used in concrete cases todetermine the bottom of the spectrum of an infinite graph.3n the other hand, finitely supported solutions of the equation e ∆ f = λf areobviously l -eigenfunctions and, therefore, they can only exist for eigenvalues λ ≥ e λ ( G ). Existence of finitely supported eigenfunctions in Penrose tilings wasfirst observed in [KS]. Their existence is a purely discrete phenomenon, since inthe case of a non-compact, connected Riemannian manifold the eigenvalue equa-tion ∆ f = λf cannot have compactly supported eigenfunctions (a fact whichis known as the unique continuation principle ; see [Ar]). These finitely sup-ported eigenfunctions coincide with the discontinuities of the integrated densityof states (or spectral density function). See, e.g., the articles [KLS, LV] and thereferences therein for more details about this connection. Examples. (a) We consider the periodic tessellation G = ( V , E , F ) in Figure1. We assume that all edges are straight Euclidean segments of length one. PSfrag replacements v Figure 1: Plane tessellation with regular triangles and hexagons
We first show that µ ( G ) = 0 : Choose a fixed radius < r < / . Then allEuclidean balls of radius r centered at all vertices in V are pairwise disjoint. Onthe other hand, the vertices in the combinatorial ball B n ( v ) are contained in theEuclidean ball of radius n , centered at v . Both facts together imply that combi-natorial balls grow only polynomially and the exponential growth is zero. As aconsequence, this graph cannot contain a binary tree as a subgraph. Moreover,using (3) , we conclude that e λ ( G ) = e λ ess ( G ) = 0 and e α ( G ) = α ( G ) = 0 . Finally, G does admit finitely supported eigenfunctions, namely, choose p ∈ R to be the center of a hexagon and define f ( p + e πi/ ) = ( − i (i.e., choose al-ternating values , − , , − , , − clockwise around the vertices of the hexagon)and f ( v ) = 0 for all other vertices. Then we have e ∆ f = f .(b) Let T p denote the p -regular tree. In this case, spectrum and essentialspectrum of the combinatorial Laplacian coincide and are given by the interval(see, e.g., [Sun, App. 3]) (cid:20) − √ p − p , √ p − p (cid:21) . onsequently, e ∆ f = λf admits a positive solution if and only if λ ≤ − √ p − /p . Moreover, we have e α ( T p ) = p − p , α ( T p ) = p − and µ ( T p ) =log( p − . Note that a regular tree doesn’t admit l -eigenfunctions. For oth-erwise, we could choose a vertex v at which our eigenfunction doesn’t vanishand take its radialisation with respect to this vertex. This radialisation would beagain a non-vanishing l -eigenfunction with the same eigenvalue and, since itsvalues would only depend on the distance to v , there would be an easy recursionformula for its values. The precise form of the recursion formula would thencontradict to the requirement that the function lies in l . Acknowledgements.
Matthias Keller would like to thank Daniel Lenzwho encouraged him to study the connection between curvature and spectraltheory. Matthias Keller was supported during this work by the German BusinessFoundation (sdw).
In the first two subsections, we provide the notions which haven’t yet beenintroduced in full detail in the Introduction. In Subsections 2.3 and 2.4, westate our main results.
Let G = ( V , E ) be a planar graph (with V and E the set of vertices and edges)embedded in R . The faces f of G are the closures of the connected componentsin R \ S e ∈ E e . The set of faces is denoted by F .We further assume that G has no loops, no multiple edges and no verticesof degree one (terminal vertices). We write e = vw , if the edge e connects thevertices v, w . Moreover, we assume that every vertex has finite degree and thatevery bounded open set in R meets only finitely many faces of G . We calla planar graph with these properties simple . The boundary of a face f is thesubgraph ∂f = ( V ∩ f, E ∩ f ). We call a sequence of edges e , . . . , e n a walk oflength n if there is a corresponding sequence of vertices v , . . . , v n +1 such that e i = v i v i +1 . A walk is called a path if there is no repetition in the correspondingsequence of vertices v , . . . , v n .A simple planar graph G is called a locally tessellating planar graph if thefollowing additional conditions are satisfied:i.) Any edge is contained in precisely two different faces.ii.) Any two faces are either disjoint or have precisely a vertex or a path ofedges in common. In the case that the length of the path is greater thenone, then both faces are unbounded.iii.) Any face is homeomorphic to the closure of an open disc D ⊂ R , to R \ D or to the upper half plane R × R + ⊂ R and its boundary is a path.Note that these properties force the graph G to be connected. Examples aretessellations R introduced in [BP1, BP2], trees in R , and particular finitetessellations on the sphere mapped to R via stereographic projection.5hen we consider the vertex degree as a function on V we write deg( v ) = | v | for v ∈ V . Moreover we define the degree | f | of a face f ∈ F to be the lengthof the shortest closed walk in the subgraph ∂f meeting all its vertices. If thereis no such finite walk we set | f | = ∞ . v ∼ w means that d ( v, w ) = 1, i.e., v and w are neighbors. A (finite or infinite) path with associated vertex sequence . . . v i v i +1 v i +2 . . . is called a geodesic , if we have d ( v i , v j ) = | i − j | for all pairsof vertices in the path. Let G = ( V , E , F ) be a locally tessellating planar graph. The operators ∆ and e ∆ were already introduced in (1) and (2). They are symmetric operators andinitially defined on the space c c ( V ) := { ϕ : V→ R | | supp ϕ | < ∞} of functions with finite support. However, they have unique self-adjoint exten-sions on different l -spaces: Let g : V → (0 , ∞ ) be a weight function on thevertices of the graph G and l ( V , g ) := { ϕ : V→ R | h ϕ, ϕ i g := X v ∈ V g ( v ) | ϕ ( v ) | < ∞} . For g = 1 we simply write l ( V ).Then the combinatorial Laplacian can be extended to a bounded self-adjointoperator on all of l ( V , deg). The physical Laplacian has also a unique self-adjoint extension in the space l ( V ) (see [We] or [Woj]). Note, however, thatthe adjacency operator need not be essentially self adjoint (see [MW, Section3] and the references therein). We denote the self-adjoint extensions of bothLaplacians, again, by e ∆ and ∆.Furthermore, we define the restriction of the combinatorial Laplacian on thecomplement of a finite set K of vertices. Let P K : l ( V , deg) → l ( V \ K, deg) bethe canonical projection and i K : l ( V \ K, deg) → l ( V , deg) be its dual operator,which is the continuation by 0 on K . We write e ∆ K = P K e ∆ i K . Of particularimportance is the bottom of the spectrum e λ ( G ) and of the essential spectrum e λ ess ( G ). e λ ( G ) can be characterised as the infimum of the Rayleight-Ritz quo-tient over all non-zero functions f ∈ l ( V , deg), i.e., e λ ( G ) = inf ( h e ∆ f, f i deg h f, f i deg : f = 0 , f ∈ l ( V , deg) ) . Similarly, e λ ess ( G ) can be obtained via e λ ess ( G ) = lim n →∞ inf ( h e ∆ B n f, f i deg h f, f i deg : f = 0 , f ∈ l ( V\ B n , deg) ) , (4)where B n are balls of radius n around any fixed vertex v ∈ V . A proof of (4)can be found in [Ke]. Obviously, we have e λ ( G ) ≤ e λ ess ( G ). Equality holds inthe following case: 6 roposition 1. Assume that there is a subgroup Γ of the automorphism groupof G with sup γ ∈ Γ d ( v, γv ) = ∞ for some vertex v ∈ V . Then we have e λ ( G ) = e λ ess ( G ) . Proof.
For the bottom of the spectrum not to lie in the essential spectrum wouldmean that it is an isolated eigenvalue of finite multiplicity. But this cannot bethe case (see Fact 1 in [Sun, p. 259]).Analogous statements hold for the bottom of the (essential) spectrum of thephysical Laplacian.
The physical and combinatorial Cheeger constants were introduced in Definition2. It is easy to see that they are linked to the physical and combinatorialLaplacians via the equations: α ( G ) = inf W ⊆ V , | W | < ∞ h ∆ χ W , χ W ih χ W , χ W i and e α ( G ) = inf W ⊆ V , | W | < ∞ h e ∆ χ W , χ W i deg h χ W , χ W i deg , where χ W denotes the characteristic function of the set W ⊆ V . Note, inparticular, that the combinatorial Cheeger constant is always bounded fromabove by e α ( G ) ≤ Theorem 1.
Let G = ( V , E , F ) be a locally tessellating planar graph and ≤ q ≤ ∞} such that | f | ≤ q for all faces f ∈ F .(a) For some a > , let κ ( v ) ≤ − a for all v ∈ V . Then we have α ( G ) ≥ qq − a. (b) For some c > , let | v | κ ( v ) ≤ − c for all v ∈ V . Then we have e α ( G ) ≥ qq − c. Moreover, the above estimates are sharp in the case of regular trees. (Note thatin the case q = ∞ we set qq − = 2 .) Remark.
The combinatorial Cheeger constant of all non-positively curved regu-lar plane tessellation G p,q (with all vertices satisfying | v | = p and faces satisfying | f | = q ) was explicitly calculated in [HJL] and [HiShi] as e α ( G p,q ) = p − p s − p − q − . Our estimate gives in this case e α ( G p,q ) ≥ ( p − q − − p ( q − . G = ( V , E , F ), let us first introduce the cut locus Cut( v ) of a vertex v ∈ V . Cut( v ) denotes the set of all vertices w , at which d v := d ( v, · ) attainsa local maximum, i.e., we have w ∈ Cut( v ) if d v ( w ′ ) ≤ d v ( w ) for all w ′ ∼ w . G is without cut locus if Cut( v ) = ∅ for all v ∈ V . Obviously, the cut locusof a finite graph is never empty. It was proved in [BP2, Thm. 1] that planetessellations with everywhere non-positive corner curvature are graphs withoutcut locus. Moreover, let T p denote the regular tree with | v | = p for all vertices. Theorem 2.
Let G = ( V , E , F ) be a locally tessellating planar graph withoutcut locus.(a) If there exists p ≥ such that | v | ≤ p ∀ v ∈ V , (5) then we have µ ( G ) ≤ µ ( T p ) = log( p − . (b) If there exist p ≥ such that (5) is satisfied and q ∈ { , , } such that | f | = q ∀ f ∈ F , (i.e., G is face-regular) then we have µ ( G ) ≤ µ ( G p,q ) = log p − q − s(cid:18) p − q − (cid:19) − . Remark.
For the reader’s convenience, Theorem 2(b) was stated in “more fa-miliar” terms of vertex and face degrees. However, the statement has an equiv-alent reformulation in terms of curvature: Let G be a locally tessellating planargraph without cut locus satisfying | f | = q for all faces and q ∈ { , , } . Forsome b ≥ , let − b ≤ κ ( v ) for all v ∈ V . Then we have µ ( G ) ≤ log( τ + p τ − , where τ = 1+ qq − b ≥ . The inequality is sharp (with the optimal choice of b ) inthe case of regular graphs G p,q . In fact, the proof will be given for this equivalentreformulation. (Note that the constants p and b in the two formulations arerelated by b = q − q p − .) Since the regular plane tessellations G p,q can be considered as combinato-rial analogues of constant curvature space forms in Riemannian geometry, it isnatural to conjecture the following discrete version of a Bishop volume compar-ison result (see, e.g., [GaHuLa, Theorem 3.101] for the case of a Riemannianmanifold).
Conjecture.
Let p, q ≥ with /p + 1 /q ≤ / be given. Then we have µ ( G ) ≤ µ ( G p,q ) , (6) for all locally tessellating planar graphs G = ( V , E , F ) without cut locus satisfying | v | ≤ p , | f | ≤ q . q = 3 and q = ∞ . However,it seems difficult to prove this seemingly obvious estimate (6) for general facedegree bounds q ≥
3. Assuming the above conjecture to be true, the comparisonof the exponential growth of a locally tessellating planar graph with upper vertexdegree bound p and of the regular tree T p , as given in Theorem 2(a), is quite goodif all faces of G satisfy | f | ≥
6. For example, we have in the case ( p, q ) = (5 , . · · · = log(2 + √
3) = µ ( G , ) ≤ µ ( T ) = log 4 = 1 . . . . . An direct consequence of [BP1, Corollary 5.2] is the following lower boundfor the exponential growth:
Theorem 3.
Let G = ( V , E , F ) be a locally tessellating planar graph withoutcut locus and a > such that κ ( v ) ≤ − a for all vertices v ∈ V . Assume there is ≤ q ≤ ∞ such that we have | f | ≤ q for all faces f ∈ F . Then we have µ ( G ) ≥ log (cid:18) qq − a (cid:19) . Moreover, this estimate is sharp in the case of regular trees. (In the case q = ∞ ,we set qq − = 2 .) We like to finish this subsection by a few additional useful facts: Let S n ( v ) = { w ∈ V | d ( v, w ) = n } be the (combinatorial) sphere of radius n about v ∈ V . If there is a uniformupper bound on the vertex degree and if s n := | S n ( v ) | is a non-decreasingsequence, one easily checks that µ ( G ) = lim sup n →∞ log s n n . (7)Yet another Cheeger constant h ( G ) was considered in [BS]: h ( G ) = inf W ⊆ V , | W | < ∞ | ∂ V W || W | , where ∂ V W is the set of all vertices v ∈ V\ W which are end points of an edgein ∂ E W . In the case that µ ( G ) is presented by (7), this Cheeger constant isrelated to the exponential growth by e µ ( G ) ≥ h ( G ) , with equality in the case of regular trees. An immediate consequence of Fujiwara’s lower estimate (3) and Theorem 1 isthe following combinatorial analogue of McKean’s Theorem (see [McK] for thecase of a Riemannian manifold): 9 orollary 1 (Combinatorial version of McKean’s Theorem) . Let G = ( V , E , F ) be a locally tessellating planar graph and ≤ q ≤ ∞ such that | f | ≤ q for allfaces f ∈ F . For some c > , let | v | κ ( v ) ≤ − c for all v ∈ V . Then we have − s − (cid:18) qq − c (cid:19) ≤ e λ ( G ) . This estimate is sharp in the case of regular trees.
Combining Theorem 2(a), the curvature version of Theorem 2(b) (see theremark of the theorem) and Fujiwara’s upper estimate (3), we obtain:
Corollary 2.
Let G = ( V , E , F ) be a locally tessellating planar graph withoutcut locus.(a) If there exists p ≥ such that | v | ≤ p ∀ v ∈ V , (8) then we have e λ ess ( G ) ≤ e λ ess ( T p ) = 1 − √ p − p . (b) If there exist q ∈ { , , } with | f | = q for all f ∈ F , and b > with − b ≤ κ ( v ) for all v ∈ V , then we have e λ ess ( G ) ≤ − p τ + √ τ −
11 + τ + √ τ − , where τ = 1 + qq − b . Next we indicate implications of the above results for the spectrum of the physical Laplacian . Let λ ( G ) and λ ess ) ( G ) denote the bottom of the (essential)spectrum of the physical Laplacian ∆ and, for n ≥
0, let m n = inf w ∈ V \ B n − ( v ) | w | and M n = sup w ∈ V \ B n − ( v ) | w | , where v ∈ V is an arbitrary vertex and B − ( v ) = ∅ . Moreover let m ∞ =lim n →∞ m n and M ∞ = lim n →∞ M n . Then we have, by [Do] λ ( G ) ≥ α ( G ) M and λ ess ( G ) ≥ α ∞ ( G ) M ∞ , (9)where α ∞ ( G ) denotes the physical Cheeger constant at infinity, defined in [Ke].In general we can also estimate, as demonstrated in [Ke], m e λ ( G ) ≤ λ ( G ) ≤ M e λ ( G ) and m ∞ e λ ess ( G ) ≤ λ ess ( G ) ≤ M ∞ e λ ess ( G ) . Via this inequalities we can estimate the bottom of the (essential) spectrum ofthe physical Laplacian ∆ by the estimates of Corollary 1 and 2 for the combi-natorial Laplacian.Before we look at an explicit example, let us mention the following resultabout the absence of finitely supported eigenfunctions in the case of non-positivecorner curvature: 10 heorem 4 (see [KLPS, Theorem 4]) . Let G = ( V , E , F ) be a plane tessel-lation (in the restricted sense of [BP2]) with non-positive corner curvature inall corners. Then the combinatorial Laplacian does not admit finitely supportedeigenfunctions. Note that Theorem 4 becomes wrong if we replace “non-positive corner cur-vature” by the weaker assumption “non-positive vertex curvature”, since Ex-ample (a) of the Introduction is a graph with vanishing vertex curvature whichadmits finitely supported eigenfunctions.Let us, finally, apply the above results in an example.
Example.
We consider the regular tessellation G , . Using our geometric re-sults in this article, we obtain e α ( G , ) ≥ and µ ( G , ) = log 1 + √ ≈ . . Proposition 1 tells us that e λ ( G , ) = e λ ess ( G , ) , and with our results in thisSubsection we can conclude that e λ ( G , ) = e λ ess ( G , ) ∈ " − √ , − √ √
77 + √ ≈ [0 . , . . Using the explicit formula for the Cheeger constant in [HJL] in this particularcase, we obtain e α ( G , ) = √ ≈ . and we can shrink this interval to e λ ( G , ) = e λ ess ( G , ) ∈ " − r , − √ √
77 + √ ≈ [0 . , . . Note that the physical Laplacian is just a multiple of the combinatorial Laplacian( ∆ = 6 e ∆ ). Finally, Theorem 4 guarantees that there are no finitely supportedeigenfunctions in G , . The heart of the proof of Theorem 1 is Proposition 2 below. An earlier versionof this proposition in the dual setting (see [BP1, Prop. 2.1]) was originallyobtained by helpful discussions with Harm Derksen. Let us first introduce someimportant notions related to a locally tessellating planar graph G = ( V , E , F ).For a finite set W ⊆ V let G W = ( W, E W , F W ) be the subgraph of G inducedby W , where E W are the edges in E with both end points in W and F W arethe faces induced by the graph ( W, E W ). Euler’s formula states for a finite andconnected subgraph G W (observe that F W contains also the unbounded face): | W | − |E W | + |F W | = 2 . (10)By ∂ F W , we denote the set of faces in F which contain an edge of ∂ E W .Moreover, we define the inner degree of a face f ∈ ∂ F W by | f | iW = | f ∩ W | .
11n the following, we need the two important formulas which hold for arbitraryfinite and connected subgraphs G W = ( W, E W , F W ). The first formula is easyto see and reads as X v ∈ W | v | = 2 |E W | + | ∂ E W | . (11)Since W is finite, the set F W contains at least one face which is not in F ,namely the unbounded face surrounding G W , but there can be more. Define C ( W ) = |F W | − |F W ∩ F| ≥
1. Note that |F W ∩ F| is the number of faces in F which are entirely enclosed by edges of E W . Sorting the following sum oververtices according to faces gives the second formula X v ∈ W X f ∋ v | f | = |F W ∩ F| + X f ∈ ∂ F W | f | iW | f | = |F W | − C ( W ) + X f ∈ ∂ F W | f | iW | f | . (12) Proposition 2.
Let G = ( V , E , F ) be a locally tessellating planar graph and W ⊂ V be a finite set of vertices such that the induced subgraph G W is connected.Then we have κ ( W ) = 2 − C ( W ) − | ∂ E W | X f ∈ ∂ F W | f | iW | f | Proof.
By the equations (11), (12) and (10) we conclude κ ( W ) = X v ∈ W − | v | X f ∋ v | f | = | W | − |E W | − | ∂ E W | |F W | − C ( W ) + X f ∈ ∂ F W | f | iW | f | = 2 − C ( W ) − | ∂ E W | X f ∈ ∂ F W | f | iW | f | . Proposition 3.
Let G = ( V, E, F ) be a locally tessellating planar graph and ≤ q ≤ ∞ such that | f | ≤ q for f ∈ F . Let W ⊂ V be a finite set of verticessuch that the induced subgraph G W is connected. Then we have | ∂ E W | ≥ qq − − C ( W ) − κ ( W )) . Proof.
Since G is locally tessellating, every edge e ∈ ∂ E W separates preciselytwo different faces. The edge obtains a direction by its start vertex to be in V\ W and its end vertex to be in W . Thus it makes sense to refer to the facesat the left and right side of the edge e . Thus every edge e ∈ ∂ E W determines aunique corner ( v, f ) ∈ W × ∂ F W , where v ∈ W is the end vertex of e and f is12he face at the left side of e . The so defined map ∂ E W → W × ∂ F W is clearlyinjective, and thus we have X f ∈ ∂ F W | f | iW = |{ ( v, f ) ∈ W × ∂ F W : v ∈ f }| ≥ | ∂ E W | . Using this fact and | f | ≤ q for all f ∈ F , we conclude with Proposition 22 − C ( W ) − κ ( W ) = | ∂ E W | − X f ∈ ∂ F W | f | iW | f | ≤ | ∂ E W | (cid:18) − q (cid:19) , which proves the inequality in the proposition.Note that the Cheeger constants in Definition 2 are obtained by taking theinfimum of a particular expression over all finite subsets W ⊂ V . In fact, wecan restrict ourselves to consider only finite sets W for which the induces graph G W is connected. This follows from the observation that, for a given finite set W ⊂ V , we can always find a non-empty subset W ⊂ W such that G W isa connected component of G w and that | ∂ E W | / vol( W ) ≤ | ∂ e W | / vol( W ) or | ∂ E W | / | W | ≤ | ∂ e W | / | W | , respectively. We can reduce the sets under consid-eration even further. Let W ⊂ V be a finite set such that G W is connected. Notethat G w has only one unbounded face. By adding all vertices of V containedin the union of all bounded faces of G w , we obtain a bigger finite set P W ⊃ W such that C ( P W ) = 1. (Note that all bounded faces of G P W are also faces ofthe original graph G .) We call a finite set P ⊂ V with connected graph G P and C ( P ) = 1 a polygon . Clearly, we have | ∂ E P W | / vol( P W ) ≤ | ∂ e W | / vol( W ) and | ∂ E P W | / | P W | ≤ | ∂ e W | / | W | . Thus it suffices for the definition of the Cheegerconstants to take the infimum only over all polygons.With this final observation we can now prove Theorem 1. Proof of Theorem 1.
Let W ⊂ V be a polygon. Since C ( W ) = 1, we concludefrom Proposition 3 that | ∂ E W || W | ≥ qq − − κ ( W ) | W | ≥ qq − a. Taking the infimum over all polygons yields part (a) of the theorem.For the proof of part (b), recall that − κ ( v ) ≥ c · | v | for all vertices v ∈ V .This implies that − κ ( W )vol( W ) = − P v ∈ W κ ( v ) P v ∈ W | v | ≥ c, and, consequently, for polygons W ⊂ V , | ∂ E W | vol( W ) ≥ qq − − κ ( W )vol( W ) ≥ qq − c. The statement follows now again by taking the infimum over all polygons.13
Proof of Theorem 2
Parts (a) and (b) of Theorem 2 have very different proofs. We present themseparately.
Proof of Theorem 2 (a).
We choose a vertex v ∈ V and introduce the followingfunctions m, M : F → { , , , . . . , ∞} : m ( f ) = min { d ( w, v ) | w ∈ ∂f } ,M ( f ) = max { d ( w, v ) | w ∈ ∂f } . Note that the face f “opens up” at distance m ( f ) and “closes up” at distance M ( f ) from v . We call a face f finite , if M ( f ) < ∞ .The idea of the proof is to “open up” successively every finite face f ∈ F intoan infinigon without violating the vertex bound. In this way, we will build up acomparison tree T with the same vertex bound p and satisfying µ ( G ) ≤ µ ( T ). Itturns out, however, that finite faces f with more than one vertex in the sphere S M ( f ) ( v ) cause problems in this “opening up” procedure (since the distancerelations to the vertex v will be changed). Therefore, we first modify thetessellation G by removing all edges connecting two vertices v, w at the samedistance to v . The modified planar graph is denoted by G = ( V , E , F ). Tokeep track, we add at each of the vertices v, w a short terminal edge. Theseterminal edges do not belong “officially” to the graph G and serve merelyas reminders that an edge can be added in their place without violating thevertex bound of the graph. Moreover, we can only guarantee µ ( G ) ≥ µ ( G ), ifthese inofficial edges are included in G . (At the end of the procedure we willreplace all “inofficial” terminal edges by infinite trees rooted in v and w .) Themodification G → G is illustrated in Figure 2. (For convenience, the verticesbelonging to distance spheres S n ( v ) are arranged to lie on concentric Euclideancircles around v .)PSfrag replacements G G v v Figure 2: Removing edges between vertices on the same spheres and replacingthem by “inofficial” terminal edges 14ote that none of the distance relations of the vertices in G (without theinofficial terminal edges) to the vertex v are changed and that we still haveCut( v ) = ∅ . Moreover, the modified graph G (without the inofficial terminaledges) has a new set of faces F . Every finite face f of G has now even degree,since f opens up at a single vertex in the sphere S m ( f ) ( v ) and f closes up at asingle vertex in the sphere S M ( f ) ( v ).We order all finite faces f , f , f , . . . of G such that we have M ( f ) ≤ M ( f ) ≤ M ( f ) ≤ ... Next we explain the first step of our procedure, namely, how to open up f into an infinigon e f . Let n = M ( f ) ≥ w ∈ ∂f such that d ( w, v ) = n .Since C ( v ) = ∅ , we can find an infinite geodesic ray w = w, w , w , · · · ∈ V such that d ( w i , v ) = n + i . We may think of v as being the origin of the planeand of w , w , . . . as being arranged to lie on the positive vertical coordinateaxis at heights n, n + 1 , . . . with straight edges between them. Now we cutour plane along this geodesic ray, i.e., replace the ray by two parallel copies ofthe ray and thus preventing the face f from closing up at distance n . In thisway, f becomes an infinigon, which we denote by e f . (In fact, we rotationallyshrink the angle 2 π to 2 π − ǫ around v to open up a conic sector of angle ǫ containing the infinigon e f .) The procedure is illustrated in Figure 3. Note thatthe vertices w i are replaced by two copies w (1) i , w (2) i , such that w ( j ) i is connectedto w ( j ) i +1 for j = 1 , w (1) i inherits all previous neighbors of w i at one sideof the ray and w (2) i inherits all previous neighbors of w i at the other side of theray (this concerns in particular also the “inofficial” vertices). In this way weobtain a new planar graph G = ( V , E , F ).PSfrag replacements G G v v f e f Figure 3: Changing the finite face f into an infinigon e f The graph G is still connected. Note also that we have | w (1)0 | + | w (2)0 | = | w | + 1 , (13) | w (1) i | + | w (2) i | = | w i | + 2 , ∀ i ≥ . (14)After including the inofficial terminal edges in the graph G , we still have | v | ≤ p ∀ v ∈ V , and (13), (14) imply that µ ( G ) ≥ µ ( G ) ≥ µ ( G ).In the second step we carry out the same procedure with the face f ∈ F ,and obtain a new connected planar graph G = ( V , E , F ), with f ∈ F e f ∈ F . Again, after including the inofficial terminaledges, the graph G has vertex bound p and satisfies µ ( G ) ≥ µ ( G ) ≥ µ ( G ).It is now clear how to repeat the procedure. Note that for every radius n ≥ j ≥ G j , G j +1 , G j +2 , . . . remainunaltered inside the balls B n ( G k , v ). This fact guarantees that there is a well-defined limiting graph associated to the sequence G j . This limit is a connectedtree T (since all faces of T are infinigons). In T , we replace now finally theinofficial terminal edges by infinite trees, rooted at the corresponding propervertices of the tree T , with branching sequence 1 , p − , p − , p − , . . . . Theseinfinite trees can be nicely fitted into the infinigons to yield an infinite planartree T with vertex bound p and satisfying µ ( T ) ≥ µ ( G ). Since we obviously have µ ( T ) ≤ µ ( T p ) = log( p − Proof of Theorem 2 (b).
We prove the equivalent curvature version of the state-ment, given in the remark after the theorem. Since | f | = q < ∞ for all faces f , G is a tessellating plane graph in the sense of [BP1] and we have κ ( v ) = 1 − q − q | v | . Since { κ ( v ) | v ∈ V} is a discrete set and bounded from below by − b , we canassume, without loss of generality, that − b is of the form 1 − q − q p , for someinteger value p ≥
3. (In fact, p is the optimal upper bound on the vertex degreeof G .)Let S n , B n be the combinatorial spheres and balls in G with respect to areference vertex v ∈ V and s n = | S n | . Corollary 6.4 of [BP1] states that wehave s n +1 − s n = 2 qq − − κ ( B n )) . Applying this equation twice, we derive s n +2 − s n +1 + s n = − qq − κ ( S n +1 ) ≤ qq − bs n +1 . Hence we obtain the following recursion inequality s n +2 ≤ τ s n +1 − s n , s ≤ p, s = 1 , with τ = 1 + qq − b ≥
1. It is easy to see that the sequence σ n +2 = 2 τ σ n +1 − σ n , σ = p, σ = 1 , (15)is strictly increasing and dominates the sequence s n . Moreover, σ n describes thecardinality of a sphere of radius n in the regular tessellation G p,q . This impliesthat µ ( G ) ≤ µ ( G p,q ).Now, we return to the sequence σ n , as defined in (15). We first consider thecase τ >
1. The recursion formula implies that σ n = u (cid:16) τ − p τ − (cid:17) n + v (cid:16) τ + p τ − (cid:17) n , with constants u, v ∈ R chosen in such a way that the initial conditions aresatisfied. Since 0 < τ − p τ − < ,
16e conclude that v = 0, for otherwise we would have σ n →
0, contradicting tothe fact that G p,q is an infinite graph. Hence, σ n behaves asymptotically like σ n ∼ v (cid:16) τ + p τ − (cid:17) n , with a positive constant v . This, together with (7) implies that µ ( G p,q ) = lim n →∞ log σ n n = log (cid:16) τ + p τ − (cid:17) . (16)In the case τ = 1, the sequence (15) is simply given by σ n = n ( p −
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