The hyperbolicity constant of infinite circulant graphs

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Open Mathematics

Open Access

Research Article

José M. Rodríguez* and José M. Sigarreta

The hyperbolicity constant

of infinite circulant graphs

DOI 10.1515/math-2017-0061

Received June 21, 2016; accepted March 30, 2017.

Abstract:IfXis a geodesic metric space andx1; x2; x32X, ageodesic triangleT D fx1; x2; x3gis the union of the three geodesicsŒx1x2,Œx2x3andŒx3x1inX. The spaceXisı-hyperbolic.in the Gromov sense/if any side of

T is contained in aı-neighborhood of the union of the two other sides, for every geodesic triangleT inX. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

Keywords:Circulant graph, Gromov hyperbolicity, Geodesics, Infinite graphs

MSC:05C75, 05C12, 05A20

1 Introduction

Hyperbolic spaces play an important role in geometric group theory and in the geometry of negatively curved spaces (see [1–3]). The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, simply connected Riemannian manifolds of negative sectional curvature bounded away from0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory (see [1–3]).

The first works on Gromov hyperbolic spaces deal with finitely generated groups (see [3]). Initially, Gromov spaces were applied to the study of automatic groups in the science of computation (see,e.g., [4]); indeed, hyperbolic groups are strongly geodesically automatic,i.e., there is an automatic structure on the group [5].

The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in [6] that the internet topology embeds with better accuracy into a hyperbolic space than into a Euclidean space of comparable dimension (formal proofs that the distortion is related to the hyperbolicity can be found in [7]); furthermore, it is evidenced that many real networks are hyperbolic (see, e.g., [8–12]). A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers (see [13–16]). Another important application of these spaces is the study of the spread of viruses through the internet (see [17, 18]). Furthermore, hyperbolic spaces are useful in secure transmission of information on the network (see [17–19]); also to traffic flow and effective resistance of networks [20–22].

*Corresponding Author: José M. Rodríguez:Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain, E-mail: jomaro@math.uc3m.es

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In [23] it was proved the equivalence of the hyperbolicity of many negatively curved surfaces and the hyperbolicity of a graph related to it; hence, it is useful to know hyperbolicity criteria for graphs from a geometrical viewpoint. In recent years, the study of mathematical properties of Gromov hyperbolic spaces has become a topic of increasing interest in graph theory and its applications (see,e.g., [11, 17–19, 23–40] and the references therein).

If.X; d /is a metric space andWŒa; b !Xis a continuous function, we define thelengthofas

L. /WDsup

nXn

iD1

d..ti 1/; .ti//W aDt0< t1< < tnDb

o

:

We say that a curve W Œa; b ! X is ageodesicif we have L.jŒt;s/ D d..t /; .s// D jt sj for every s; t 2 Œa; b, whereLandd denote length and distance, respectively, andjŒt;sis the restriction of the curve to the intervalŒt; s(thenis equipped with an arc-length parametrization). The metric spaceXis saidgeodesicif for every couple of points inXthere exists a geodesic joining them; we denote byŒxyany geodesic joiningxand y; this notation is ambiguous, since in general we do not have uniqueness of geodesics, but it is very convenient. Consequently, any geodesic metric space is connected. If the metric spaceXis a graph, then the edge joining the verticesuandvwill be denoted byŒu; v.

Along the paper we just consider graphs with every edge of length1. In order to consider a graphGas a geodesic metric space, identify (by an isometry) any edgeŒu; v2E.G/with the intervalŒ0; 1in the real line; then the edge Œu; v(considered as a graph with just one edge) is isometric to the intervalŒ0; 1. Thus, the points inGare the vertices and, also, the points in the interior of any edge ofG. In this way, any connected graphGhas a natural distance defined on its points, induced by taking shortest paths inG, and we can seeG as a metric graph. Ifx; y are in different connected components ofG, we definedG.x; y/ D 1. Throughout this paper,G D .V; E/ D .V .G/; E.G// denotes a simple (without loops and multiple edges) graph (not necessarily connected) such that every edge has length1andV ¤ ;. These properties guarantee that any connected component of any graph is a geodesic metric space. Note that to exclude multiple edges and loops is not an important loss of generality, since [27, Theorems 8 and 10] reduce the problem of compute the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs.

IfX is a geodesic metric space andx1; x2; x32X, the union of three geodesicsŒx1x2,Œx2x3andŒx3x1is ageodesic trianglethat will be denoted byT D fx1; x2; x3gand we will say thatx1; x2andx3are the vertices ofT; it is usual to write alsoT D fŒx1x2; Œx2x3; Œx3x1g. We say thatT isı-thinif any side ofT is contained in theı-neighborhood of the union of the two other sides. We denote byı.T /the sharp thin constant ofT, i.e. ı.T /WD inffı 0 W T isı-thing:The spaceX isı-hyperbolic.or satisfies theRips conditionwith constantı/ if every geodesic triangle inX isı-thin. We denote byı.X /the sharp hyperbolicity constant ofX, i.e.ı.X / WD supfı.T / W T is a geodesic triangle in Xg:We say thatX ishyperbolicifX isı-hyperbolic for some ı 0; thenXis hyperbolic if and only ifı.X / <1:IfXhas connected componentsfXigi2I, then we defineı.X /WD supi2Iı.Xi/, and we say thatXis hyperbolic ifı.X / <1.

If we have a triangle with two identical vertices, we call it abigon; note that since this is a special case of the definition, every geodesic bigon in aı-hyperbolic space isı-thin.

In the classical references on this subject (see,e.g., [1, 2, 41]) appear several different definitions of Gromov hyperbolicity, which are equivalent in the sense that ifX isı-hyperbolic with respect to one definition, then it is ı0-hyperbolic with respect to another definition (for someı0related toı). The definition that we have chosen has a deep geometric meaning (see,e.g., [2]).

Trivially, any bounded metric space X is..diamX /=2/-hyperbolic. A normed linear space is hyperbolic if and only if it has dimension one. A geodesic space is0-hyperbolic if and only if it is a metric tree. If a complete Riemannian manifold is simply connected and its sectional curvatures satisfyK cfor some negative constantc, then it is hyperbolic. See the classical references [1, 2, 41] in order to find further results.

We want to remark that the main examples of hyperbolic graphs are the trees. In fact, the hyperbolicity constant of a geodesic metric space can be viewed as a measure of how “tree-like” the space is, since those spacesXwith ı.X /D0are precisely the metric trees. This is an interesting subject since, in many applications, one finds that the borderline between tractable and intractable cases may be the tree-like degree of the structure to be dealt with (see,

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A graph iscirculantif it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. There are large classes of circulant graphs. For instance, every cycle graph, complete graph, crown graph and Möbius ladder is a circulant graph. A complete bipartite graph is a circulant graph if and only if it has the same number of vertices on both sides of its bipartition. A connected finite graph is circulant if and only if it is the Cayley graph of a cyclic group, see [43]. Every circulant graph is a vertex transitive graph and a Cayley graph [44]. It should be noted that Paley graphs is an important class of circulant graph, which is attracting great interest in recent years (see,e.g., [45]).

The circulant is a natural generalization of the double loop network and was first considered by Wong and Coppersmith [46]. Circulant graphs are interesting by the role they play in the design of networks. In the area of computer networks, the standard topology is that of a ring network; that is, a cycle in graph theoretic terms. However, cycles have relatively large diameter, and in an attempt to reduce the diameter by adding edges, we wish to retain certain properties. In particular, we would like to retain maximum connectivity and vertex-transitivity. Hence, most of the earlier research concentrated on using the circulant graphs to build interconnection networks for distributed and parallel systems [47], [48]. The term circulant comes from the nature of its adjacency matrix. A matrix is circulant if all its rows are periodic rotations of the first one. Circulant matrices have been employed for designing binary codes [49]. Theoretical properties of circulant graphs have been studied extensively and surveyed [47].

For a finite graph withnvertices it is possible to computeı.G/in time O.n3:69/[50] (this is improved in [10, 51]). Given a Cayley graph (of a presentation with solvable word problem) there is an algorithm which allows to decide if it is hyperbolic [52]. However, deciding whether or not a general infinite graph is hyperbolic is usually very difficult. Therefore, it is interesting to relate hyperbolicity with other properties of graphs. The papers [24, 29, 40] prove, respectively, that chordal, k-chordal and edge-chordal are hyperbolic. Moreover, in [24] it is shown that hyperbolic graphs are path-chordal graphs. These results relating chordality and hyperbolicity are improved in [33]. In the same line, many researches have studied the hyperbolicity of other classes of graphs: complement of graphs [53], vertex-symmetric graphs [54], line graphs [55], bipartite and intersection graphs [56], bridged graphs [32], expanders [22], Cartesian product graphs [57], cubic graphs [58], and random graphs [37–39].

In this paper we prove that infinite circulant graphs and their complements are hyperbolic (see Theorems 2.3, 2.4 and 3.15). We obtain in Theorems 3.7 and 3.8 several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs, and the precise value of the hyperbolicity constant of many circulant graphs. Besides, Theorem 3.15 provides sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

2 Every circulant graph is hyperbolic

Let .X; dX/ and .Y; dY/ be two metric spaces. A map f W X ! Y is said to be an.˛; ˇ/-quasi-isometric

embedding, with constants˛1; ˇ0if for everyx; y2X:

˛ 1dX.x; y/ ˇdY.f .x/; f .y//˛dX.x; y/Cˇ:

The functionf is"-fullif for eachy2Y there existsx2XwithdY.f .x/; y/".

A mapf WX !Y is said to be aquasi-isometry, if there exist constants˛1; ˇ; "0such thatf is an "-full.˛; ˇ/-quasi-isometric embedding.

A fundamental property of hyperbolic spaces is the following:

Theorem 2.1(Invariance of hyperbolicity). Letf WX !Y be an.˛; ˇ/-quasi-isometric embedding between the geodesic metric spacesX andY. IfY isı-hyperbolic, thenX isı0-hyperbolic, whereı0is a constant which just depends on˛; ˇ; ı.

Besides, iff is"-full for some"0(a quasi-isometry) andXisı-hyperbolic, thenY isı0-hyperbolic, where ı0is a constant which just depends on˛; ˇ; ı; ".

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Definition 2.2. LetGbe any circulant connected infinite graph andg2Aut.G/withhgi DAut.G/. Consider the graphGwithV .G/WDV .G/andE.G/WD˚

Œgn.v0/; gnC1.v0

n2Zfor some fixedv

02V .G/. We define

n.G; g/WDdG v0; g.v0/

;

N.G; g/WDmax˚dG.v0; w/j Œv0; w2E.G/ :

Note that the definition ofn.G; g/andN.G; g/do not depend on the choice ofv0, sinceGis a circulant graph. Hence, for everyv2V .G/,

n.G; g/DdG v; g.v/

DdG v; g 1.v/

;

N.G; g/Dmax˚dG.v; w/j Œv; w2E.G/ :

Theorem 2.3. Any circulant connected infinite graphGsatisfies the inequalityı.G/ c, wherec is a constant which just depends onn.G; g/andN.G; g/.

Proof. For each n 2 Z, letwn be the midpoint of the edgeŒgn.v0/; gnC1.v0/ 2 E.G/. Let us define a map

i WG !Gas follows: for eachn2Z, leti.u/WDgn.v0/for everyu2Œwn 1wnn fwn 1g. Note thati is the inclusion map onV .G/DV .G/; hence,i is.1=2/-full.

Fix u; v 2 V .G/ D V .G/. Let u0 D u; u1; : : : ; uk 1; uk D v 2 V .G/ with dG.u; v/ D

Pk

jD1dG.uj 1; uj/andŒuj 1; uj2E.G/for every1j k; then we have

dG.u; v/ k

X

jD1

dG.uj 1; uj/ k

X

jD1

N.G; g/ dG.uj 1; uj/DN.G; g/ dG.u; v/:

Fixu; v2Gand a geodesicŒuvinG. Recall thati.u/; i.v/2V .G/DV .G/. We have

dG.u; v/dG u; i.u/CdG i.u/; i.v/CdG i.v/; v

1=2CN.G; g/ dG i.u/; i.v/C1=2 DN.G; g/ dG i.u/; i.v/C1;

1

N.G; g/dG.u; v/dG i.u/; i.v/

C 1

N.G; g/ dG i.u/; i.v/

C1:

Fixu; v2V .G/DV .G/. Letv0Du; v1; : : : ; vr 1; vr Dv 2V .G/withdG.u; v/DPjrD1dG.vj 1; vj/ andŒvj 1; vj2E.G/; thenvj Dgi.vj 1/for somei 2 f1; 1gand we have

dG.u; v/ r

X

jD1

dG.vj 1; vj/D r

X

jD1

n.G; g/ dG.vj 1; vj/Dn.G; g/ dG.u; v/:

Fixu; v2G. We have

dG i.u/; i.v/

n.G; g/ dG i.u/; i.v/

n.G; g/ dG i.u/; uCdG.u; v/CdG v; i.v/

n.G; g/ 1=2CdG.u; v/C1=2

Dn.G; g/ dG.u; v/Cn.G; g/:

We conclude thati is a.1=2/-full maxfN.G; g/; n.G; g/g; n.G; g/

-quasi-isometry and Theorem 2.1 gives the result, sinceGis0-hyperbolic.

Theorem 2.4. Every circulant graph is hyperbolic.

Proof. Let us consider any fixed circulant graphG. IfGis a finite graph, then it is.diamG/-hyperbolic. Assume now thatGis an infinite graph.

IfGis connected, then Theorem 2.3 gives that it is hyperbolic.

IfGis not connected, then it has just a finite number of isomorphic connected componentsG1; : : : ; Gr;since

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3 Bounds for the hyperbolicity constant of infinite circulant graphs

Letfa1; a2; : : : ; akgbe a set of integers such that0 < a1< < ak. We define the circulant graphC1.a1; : : : ; ak/ as the infinite graph with verticesZand such thatN.j /D fj˙aigkiD1is the set of neighbors of each vertexj2Z. ThenC1.a1; : : : ; ak/is a regular graph of degree2k. IfkD1, thenC1.1/is isometric to the Cayley graph ofZ, which is0-hyperbolic. Hence, in what follows we just consider circulant graphs withk > 1.

Denote byJ.G/the set of vertices and midpoints of edges inG, and bybtcthe lower integer part oft. The following results in [25] will be useful.

Theorem 3.1([25, Theorem 2.6]). For every hyperbolic graphG,ı.G/is a multiple of1=4.

As usual, bycyclewe mean a simple closed curve, i.e., a path with different vertices, unless the last one, which is equal to the first vertex.

Theorem 3.2([25, Theorem 2.7]). For any hyperbolic graphG, there exists a geodesic triangleT D fx; y; zgthat is a cycle withx; y; z2J.G/andı.T /Dı.G/.

We also need the following technical lemmas.

Lemma 3.3. For any integersk > 1and1 < a2< < ak, considerGDC1.1; a2; : : : ; ak/. Then the following

statements are equivalent: .1/ dG 0;

ak=2

˘

D

ak=2

˘

and, ifakis odd, thendG 0;

ak=2

˘

C1

ak=2

˘

. .2/ a2ak 1.

.3/We have eitherkD2orkD3anda2Da3 1.

Proof. Assume that.1/holds. Ifk D 2, then.2/holds; hence, we can assumek 3. Definer WD

ak=2

˘

. If a2r, thenrDdG.0; r/by hypothesis andrDdG.0; r/dG.0; a2/CdG.a2; r/1Cr a2< r, which is a contradiction. Thus we concludea2> r. Therefore,

rDdG.0; r/dG.0; a2/CdG.a2; r/1Ca2 r;

anda22r 1. Hence,a2ak 1ifakis even. Sincea2> r, ifakis odd, then

rdG.0; rC1/dG.0; a2/CdG.a2; rC1/1Ca2 .rC1/;

anda22rDak 1. Then.2/holds.

A simple computation provides the converse implication. The equivalence of.2/and.3/is elementary.

Let us define the subsetEof infinite circulant graphs asEWD˚

C1.1; 2mC1; 2mC2; 2mC3/ m1.

Lemma 3.4. Consider any integersk > 1and1 < a2< < akwitha2< ak 1, andGDC1.1; a2; : : : ; ak/… E. Then

min˚

dG.0; u/; dG.aj; u/

jak

2

k

1; (1)

for everyu2Zwith0uaj and1j k.

Proof. Sincea2< ak 1, Lemma 3.3 gives:

.i /ifakis even, thendG 0;

ak=2

˘

<

ak=2

˘

, .i i /ifakis odd, thendG 0;

ak=2

˘

<

ak=2

˘

ordG 0;

ak=2

˘

C1

<

ak=2

˘

. If dG 0;

ak=2

˘

<

ak=2

˘

, then inequality (1) trivially holds. Hence, we can assume that ak is odd,

dG 0;

ak=2

˘

D

ak=2

˘

anddG 0;

ak=2

˘

C1

<

ak=2

˘

. These facts imply thatdG 0;

ak=2

˘

C1

D

ak=2

˘

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Ifa2

ak=2

˘

, thendG 0;

ak=2

˘

<

ak=2

˘

, which is a contradiction. Therefore,a2>

ak=2

˘

and

jak

2

k

1DdG

0;jak 2

k

C11Ca2

jak

2

k

1; ak 2a2< ak 1:

We concludea2Dak 2, andkD3orkD4. IfkD4, thena2Da4 2,a3Da4 1andG2E, sinceakis odd. This is a contradiction, and we concludekD3anda2Da3 2.

Ifj D1, then min˚

dG.0; u/; dG.1; u/ D0for every0u1. Ifj D2, then for every0ua2

min˚dG.0; u/; dG.a2; u/ min

˚

u; a2 u

ja2

2

k

Dja3

2

k

1:

Ifj D 3, then dG

a3=2

˘

; a3

D dG 0;

a3=2

˘

C1

D

a3=2

˘

1, and for every0 u a3with u ¤

a3=2˘;a3=2˘C1,

min˚

dG.0; u/; dG.a3; u/ min˚u; a3 u

ja3

2

k

1:

This finishes the proof.

Proposition 3.5. Consider any integersk > 1and1 < a2 < < ak. If we have eitherk D2, ork D 3and

a2Da3 1, orC1.1; a2; : : : ; ak/2E, then

ı C1.1; a2; : : : ; ak/

1

2C

jak

2

k

:

Proof. DefinerWD

ak=2

˘

andGWDC1.1; a2; : : : ; ak/.

Assume first that we have eitherk D2, ork D3anda2Da3 1. Consider the curves1; 2inGjoining

xWD0andyWDrC.rC1/akgiven by

1WDŒ0; 1[Œ1; 2[ [Œr 1; r[Œr; rCak[ŒrCak; rC2ak[ [ŒrCrak; rC.rC1/ak;

2WDŒ0; ak[Œak; 2ak[ [Œrak; .rC1/ak[Œ.rC1/ak; .rC1/akC1[

[Œ.rC1/akC1; .rC1/akC2[ŒrC.rC1/ak 1; rC.rC1/ak:

Lemma 3.3 gives that1and2are geodesics; thendG.x; y/DL.1/DL.2/D2rC1. LetT be the geodesic bigonT D f1; 2ginG. Ifpis the midpoint ofŒr; rCak, thendG.p; x/DdG.p; y/DrC1=2and Lemma 3.3 gives thatdG.p; 2/DrC1=2. Hence,

ı C1.1; a2; : : : ; ak/

dG.p; 2/D

1 2C

jak

2

k

:

Assume now thatC1.1; a2; : : : ; ak/2E. Note thatrCrak D.rC1/ak 1, sinceakDak 1C1is odd. Consider the curves1; 2; 3inG

1WDŒ r rak; r .r 1/ak[ [Œ r 2ak; r ak[Œ r ak; r[Œ r; r

[Œr; rCak[ŒrCak; rC2ak[ [ŒrC.r 1/ak; rCrak;

2WDŒ .rC1/ak 1; rak 1[ [Œ 2ak 1; ak 1[Œ ak 1; 0;

3WDŒ0; ak 1[Œak 1; 2ak 1[ [Œrak 1; .rC1/ak 1;

joiningx WD .rC1/ak 1,y WD.rC1/ak 1andz WD0. One can check that1,2and3are geodesics in

G. LetT be the geodesic triangleT D f1; 2; 3g. Note thatdG.0; r/D r, sinceG 2 E. Therefore, ifpis the midpoint ofŒ r; r, then

ı.G/dG.p; 2[3/D

1

2CdG.r; 3/D 1

2CdG.r;f0; 2rg/D 1 2CrD

1 2C

jak

2

k

:

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Lemma 3.6. Consider any integersk > 1and1 < a2< < ak. Ifx; y2J C1.1; a2; : : : ; ak/

andxandy are not related, then

dG.x; y/1C

jak

2

k

:

Furthermore, ifa2< ak 1andC1.1; a2; : : : ; ak/…E, then

dG.x; y/

jak

2

k

:

Proof. LetGDC1.1; a2; : : : ; ak/. Ifx1y1y2x2, then the cycle

WDŒx1; x1C1[Œx1C1; x1C2[ [Œy1 1; y1[Œy1; y2[Œy2; y2C1[

[Œx2 2; x2 1[Œx2 1; x2[Œx2; x1

has length at most1Cak. Sinceis a cycle containing the pointsxandy, we have

dG.x; y/

1 2L. /

1Cak

2 1C

jak

2

k

:

If a2 < ak 1 andG … E, then Lemma 3.4 givesdG.x; y1/ 1=2Cak=2

˘

1 D

ak=2

˘

1=2and dG.x; y/

ak=2

˘

.

Ify1x1x2y2, then the same argument gives these inequalities.

Ifx1y1< x2y2, thenx2 y1Cy1 x1akand minfx2 y1; y1 x1g bak=2c. Ifx2 y1 bak=2c, thendG.x; y/ dG.x; x2/Cx2 y1CdG.y1; y/ 1C bak=2c. Ify1 x1 bak=2c, thendG.x; y/

dG.x; x1/Cy1 x1CdG.y1; y/ 1C bak=2c. Ifa2< ak 1andC1.1; a2; : : : ; ak/…E, then Lemma 3.4 also givesdG.x; y1/ak=2

˘

1=2anddG.x; y/ak=2

˘

.

Ify1x1< y2x2, then the same argument gives these inequalities.

We can state now the main result of this section, which provides a sharp upper bound for the hyperbolicity constant of infinite circulant graphs.

Theorem 3.7. For any integersk > 1and1 < a2< < ak,

ı C1.1; a2; : : : ; ak/

1

2C

jak

2

k

;

and the equality is attained if and only if we have eitherkD2, orkD3anda2Da3 1, orkD4,a2Da4 2,

a3Da4 1anda4is odd.

Proof. In order to boundı.G/withGDC1.1; a2; : : : ; ak/, let us consider a geodesic triangleT D fx; y; zginG andp2Œxy; by Theorem 3.2 we can assume thatT is a cycle withx; y; z2J.G/. We consider several cases.

.A/Ifxandyare not related, then Lemma 3.6 gives

dG.p; Œxz[Œyz/dG.p;fx; yg/

1

2dG.x; y/ 1 2C

1 2

jak

2

k

: (2)

.B/Assume thatxR y. Without loss of generality we can assume thatxL y. Denote byw1; : : : ; wmthe vertices in

Œxysuch thatw12 fx1; x2g,wm2 fy1; y2ganddG.wj; wjC1/D1for every1j < m; we define

i0WDminf1i mjwj x2 8i j mg; x0WDwi0:

.B:1/Assume that eitherx2V .G/orxis the midpoint of an edgeŒr; rC12E.G/for somer2Z. Ifx0Dx2, thenL.Œxx0/1=2. Ifx0> x2, then the cycle

WDŒwi0; wi0 1[Œwi0 1; wi0 1C1[Œwi0 1C1; wi0 1C2[ [Œwi0 2; wi0 1[Œwi0 1; wi0

has length at most1Cak. Sincex2V .G/orxis the midpoint of an edgeŒr; rC1, thusis a cycle containing the pointsxandx0, and we have

L.Œxx0/DdG.x; x0/

1 2L. /

1Cak

2 < 3 2C

jak

2

k

(8)

.B:2/Assume now thatxis the midpoint of an edgeŒr; rCaj2E.G/for somer2Zand1 < jk. Ifx0Dx2, thenL.Œxx0/ D1=2. Ifwi0 1 Dx1, thenL.Œxx0/ D3=2. Ifx0 > x2andwi0 1 ¤x1, then we have either wi0 1< x1< x2< wi0orx1< wi0 1< x2< wi0. In the first case the cycle

WDŒwi0; wi0 1[Œwi0 1; wi0 1C1[ [Œx1 1; x1[Œx1; x2[Œx2; x2C1[ [Œwi0 1; wi0

has length at mostak. Sinceis a cycle containing the pointsxandx0, we have

L.Œxx0/DdG.x; x0/

1 2L. /

ak

2 :

Assume thatx1< wi0 1< x2< wi0. Then

L.Œxx0/DdG.x; x0/

1

2CminfdG.x2; wi0 1/; dG.wi0 1; x1/g C1

3

2Cminfx2 wi0 1; wi0 1 x1g 3 2C

jak

2

k

:

Therefore, in Case.B/, ifp2Œxx0nB.x0; 3=4/, then

dG.p; Œxz[Œyz/dG.p; x/L Œxx0nB.x0; 3=4/

DL Œxx0

3

4

3 4C

jak

2

k

: (3)

Define

j0WDmaxf1i mjwj y1 81j ig; y0WDwj0:

A similar argument to the previous one shows that ifp2Œy0ynB.y0; 3=4/, then

dG.p; Œxz[Œyz/

3 4C

jak

2

k

: (4)

SinceT is a continuous curve, we have

Œxz[Œyz

\ fx1; x2g ¤ ;; Œxz[Œyz\ fy1; y2g ¤ ;:

.a/Assume thaty02Œxx0n fx0g.

.a:1/IfdG.x0; y0/2, then

L.Œxy/DL.Œxx0/CL.Œy0y/ L.Œy0x0/

3 2C

jak

2

k

C3

2C

jak

2

k

2D1C2

jak

2

k

;

dG.p; Œxz[Œyz/dG.p;fx; yg/

1 2L Œxy

1

2C

jak

2

k

:

.a:2/IfdG.x0; y0/ D1, thenwj0 D y0 D wi0 1,wi0 D x0 D wj0C1and the definitions ofx0andy0give y0< x2x0andy0y1< x0. SinceŒx0; y02E.G/,

dG.x2; y1/

1Cak

2 ;

L Œxy

DdG.x; y/dG.x; x2/CdG.x2; y1/CdG.y1; y/

1 2C

1Cak

2 C

1 2 D

3Cak

2 ;

dG.p; Œxz[Œyz/dG.p;fx; yg/

1 2L Œxy

3Cak

4

1 2C

jak

2

k

:

.b/Assume thaty0…Œxx0n fx0g. Sincex2x0; y0y1andŒxz[Œyzis a continuous curve joiningxandy, ifp 2V .G/\Œx0y0Œxy, then there existu; v2V .G/\ Œxz[ŒyzwithŒu; v2E.G/andup v. SinceT is a cycle, we have

dG.p; Œxz[Œyz/dG.p;fu; vg/minfp u; v pg

jak

2

k

:

Therefore, ifp2Œx0y0[B.x0; 3=4/[B.y0; 3=4/, then

dG.p; Œxz[Œyz/ <

3 4C

jak

2

k

(9)

This inequality, (2), (3) and (4) give

ı C1.1; a2; : : : ; ak/

3

4C

jak

2

k

: (5)

By Theorem 3.1, in order to finish the proof it suffices to show that the equality in (5) is not attained. Seeking for a contradiction, assume that the equality is attained. The proof of (5) gives that we have

dG.p; Œxz[Œyz/D

3 4C

jak

2

k

;

wherepis the point inŒxx0withdG.p; x0/ D3=4or the point inŒy0ywithdG.p; y0/ D3=4. By symmetry, without loss of generality we can assume thatpis the point inŒxx0withdG.p; x0/D3=4; thusdG.p; wi0 1/D 1=4. Therefore, we are in Case.B:2/withx1< wi0 1< x2< wi0and

L.Œxx0/D

3

2CminfdG.x2; wi0 1/; dG.wi0 1; x1/g D 3

2Cminfx2 wi0 1; wi0 1 x1g D 3 2C

jak

2

k

:

Then

minfdG.x2; wi0 1/; dG.wi0 1; x1/g Dminfx2 wi0 1; wi0 1 x1g D

jak

2

k

;

and we conclude

jak

2

k

x2 wi0 1; wi0 1 x1

jak

2

k

C1: (6)

.I /Assume thatx22Œxx0. SinceT is a cycle, thenx12Œxz[Œyzandx1< wi0 1. Hence, sinceŒxz[Œyzis a continuous curve joiningxandy, anddG.p; wi0 1/D1=4, we obtain as above

3 4C

jak

2

k

DdG.p; Œxz[Œyz/dG.p; wi0 1/CdG.wi0 1; Œxz[Œyz/ 1 4C

jak

2

k

;

which is a contradiction.

.II /Assume thatx12Œxx0.

.II:1/Ifx2 x1Dx0 wi0 1Dak, thenwi0 1 x1Dx0 x2,dG.x1; wi0 1/DdG.x2; x0/and

dG.x; x0/DdG.x; x1/CdG.x1; wi0 1/CdG.wi0 1; x0/ > dG.x; x1/CdG.x1; wi0 1/

D 1

2CdG.x2; x0/DdG.x; x2/CdG.x2; x0/dG.x; x0/;

which is a contradiction.

.II:2/Ifx2 x1< ak, then (6) gives

x2 wi0 1Dwi0 1 x1D

jak

2 k ; and 3 4C

jak

2

k

DdG.p; Œxz[Œyz/dG.p; x2/dG.p; wi0 1/CdG.wi0 1; x2/ 1 4C

jak

2

k

;

which is a contradiction.

.II:3/Assumex2 x1Dakandx0 wi0 1< ak. Sincex12Œxx0, we havewi0 1 x1D

ak=2

˘

. .II:3:1/Ifx2 wi0 1D

ak=2

˘

, then

3 4C

jak

2

k

DdG.p; Œxz[Œyz/dG.p; x2/dG.p; wi0 1/CdG.wi0 1; x2/ 1 4C

jak

2

k

;

which is a contradiction.

.II:3:2/Ifx2 wi0 1D

ak=2

˘

C1, thenak Dx2 x1D2

ak=2

˘

C1and

dG.x0; x2/x0 x2Dx0 wi0 1 x2 wi0 1

< ak

jak

2

k

1Djak

2

k

;

3 4 C

jak

2

k

DdG.p; Œxz[Œyz/dG.p; x2/dG.p; x0/CdG.x0; x2/ <

3 4C

jak

2

k

(10)

which is a contradiction.

This finishes the proof of the inequality.

If we have eitherkD2, orkD3anda2Da3 1, orkD4,a2Da4 2,a3Da4 1anda4is odd, then Proposition 3.5 gives

ı C1.1; a2; : : : ; ak/

1

2C

jak

2

k

:

Since we have the converse inequality, we conclude that the equality holds.

Assume now that the equality holds. Denote byGthe circulant graphC1.1; a2; : : : ; ak/. By Theorem 3.2 there exist a geodesic triangleT D fx; y; zginGthat is a cycle withx; y; z 2J.G/;andp 2ŒxywithdG.p; Œxz[

Œyz/D1=2C

ak=2

˘

. Hence,p2J.G/.

Seeking for a contradiction, assume thata2< ak 1and thatG…E. If x and y are not related, then Lemma 3.6 gives dG.x; y/

ak=2

˘

. Since dG.x; y/ D dG.x; p/C

dG.p; y/ 1C2

ak=2

˘

, thusxandy are related, and without loss of generality we can assumexL y. Since dG.p; x/; dG.p; y/1=2C

ak=2

˘

, the previous argument gives thatxandpare related, andpandyare related. .i /Assume firstxL pandpL y.

Sincep1 x1 p1 x2 dG.x2; p1/

ak=2

˘

1=2, we havep1 x1 p1 x2

ak=2

˘

, and there is some pointu 2 .Œxz[Œyz/\V .G/with u p1. A similar argument gives that there is some point

v2.Œxz[Œyz/\V .G/withp2v. Denote byuandvany couple of vertices satisfying these properties. We are going to prove thatv u > ak.

Ifp2V .G/, then

min˚p u; v p min˚dG.u; p/; dG.p; v/

1 2C

jak

2

k

;

min˚

p u; v p 1Cjak

2

k

; v u2C2

jak

2

k

> ak:

Ifpis the midpoint of some edgeŒm; mCajwithm 2Zand2 j k, then Lemma 3.4 givesdG.p; w/

ak=2

˘

1=2 < 1=2C

ak=2

˘

for everyw2ZwithmwmCaj, sincea2< ak 1andG…E. Furthermore,

min˚m u; v m aj min

˚

dG.u; m/; dG.mCaj; v/

jak

2

k

;

v uajC2

jak

2

k

2C2

jak

2

k

> ak:

Finally, assume that p is the midpoint of some edge Œm; mC 1 with m 2 Z. By Lemma 3.3, we have

dG 0;ak=2

˘

ak=2

˘

1ordG 0;ak=2

˘

C1

ak=2

˘

1. Therefore,

dG

p; p˙jak

2

k

C1

2

jak

2

k 1

2; min

˚

p u; v p 3

2C

jak

2

k

;

v u3C2jak 2

k

2C2jak 2

k

> ak:

Hence, we have provedv u > akin every case.

Since ifpis the midpoint of some edgeŒm; mCajwithm2Zand2j kthendG.p; w/ < 1=2C

ak=2

˘

for everyw 2 Zwith m w mCaj, and Œxz[Œyzis a continuous curve joining u andv, there exist

u0; v02.Œxz[Œyz/\V .G/withu0p1,p2v0andŒu0; v02E.G/. Hence,akv0 u0> ak, which is a contradiction.

.i i / Assume now that pL x or yL p. By symmetry, we can assume that pL x and thus p 2 Œxx0 and

dG.p; x0/1. SincedG.x; x0/3=2Cak=2

˘

, we have1=2C

ak=2

˘

DdG p; Œxz[ŒyzdG.x; p/

1=2C

ak=2

˘

. Therefore,dG.x; p/ D 1=2C

ak=2

˘

,dG.p; x0/ D 1,p 2 V .G/andx is the midpoint of

Œx1; x22E.G/. ThusdG.p; x1/ak=2

˘

,dG.p; x2/ak=2

˘

andpx1 x2x0. By Lemma 3.3, we havedG 0;

ak=2

˘

ak=2

˘

1ordG 0;

ak=2

˘

C1

ak=2

˘

1. IfdG 0;ak=2

˘

ak=2

˘

1, thenx1ak=2

˘

C1,x2ak

ak=2

˘

1

and

1x2 x1ak

ak=2

˘

1

ak=2

˘

1Dak 2

ak=2

˘

0;

(11)

IfdG 0;

ak=2

˘

C1

ak=2

˘

1, thenx2

ak=2

˘

,x1

ak=2

˘

and1x2 x1

ak=2

˘

ak=2

˘

0, which is a contradiction.

Therefore, we conclude in every case thata2ak 1orG 2E. Hence, Lemma 3.3 giveskD2, orkD3 anda2Da3 1, orkD4,a2Da4 2,a3Da4 1anda4is odd.

We also have a sharp lower bound for the hyperbolicity constant.

Theorem 3.8. For any integersk > 1and1 < a2< < akwe have

ı C1.1; a2; : : : ; ak/

3

2;

and the equality is attained ifak Dk.

Proof. DefineGDC1.1; a2; : : : ; ak/, and consider the geodesics1; 2inGgiven by

1WDŒ0; ak[Œak; 2ak[Œ2ak; 2akC1;

2WDŒ0; 1[Œ1; 1Cak[Œ1Cak; 1C2ak:

LetT be the geodesic bigonT D f1; 2ginG. Ifpis the midpoint ofŒak; 2ak, then

ı.G/dG.p; 2/Dmin

n

dG.p; ak/CdG.ak; 1Cak/; dG.p; 2ak/CdG.2ak; 2akC1/;

1 2L.1/

o

D 3

2:

Assume now thatak D k, i.e.,G D C1.1; a2; : : : ; ak/ D C1.1; 2; : : : ; k/. Therefore,dG.m; mCw/ D 1 for everym; w 2 Zwithjwj k. Note that ifx; y 2 J.G/andx andy are not related, thenjx1 y1j < k,

dG.x1; y1/D1and

dG.x; y/dG.x; x1/CdG.x1; y1/CdG.y1; y/D2: (7)

Let us consider a geodesic triangleT D fx; y; zginGandp 2 Œxy; by Theorem 3.2 we can assume thatT is a cycle withx; y; z2J.G/. We consider several cases.

.A/Ifxandyare not related, then (7) gives

dG.p; Œxz[Œyz/dG.p;fx; yg/

1

2dG.x; y/1:

.B/Assume thatxR y. Without loss of generality we can assume thatxL y. Definewj,i0,x0andy0as in the proof of Theorem 3.7. Thenwi0 1< x2wi0 Dx0.

Ifx0Dx2, thenL.Œxx0/1=2. Ifx0> x2, thenwi0 1< x2< wi0. Sincex2 wi0 wi0 wi0 1k, dG.x2; x0/D1and

L.Œxx0/DdG.x; x0/dG.x; x2/CdG.x2; x0/

1 2C1D

3 2:

Therefore, in both cases, ifp2Œxx0, then

dG.p; Œxz[Œyz/dG.p; x/L Œxx0

3

2:

A similar argument to the previous one shows that ifp2Œy0y, thendG.p; Œxz[Œyz/3=2. Ify02Œxx0, then

dG.p; Œxz[Œyz/3=2holds for everyp2Œxy. Consider now the casey0…Œxx0.

Sincex2x0andy0y1, everyp2V .G/\Œx0y0Œxyverifiesx2py1. SinceT is a continuous curve, we obtain

Œxz[Œyz

\ fx1; x2g ¤ ;; Œxz[Œyz

\ fy1; y2g ¤ ;:

SinceŒxz[Œyzis a continuous curve joiningxandy, ifp 2 V .G/\Œx0y0, then there existu; v 2 V .G/\

Œxz[Œyz

withŒu; v2E.G/andupv. Hence,p uv uk,v pv ukand

(12)

Therefore, ifp2Œx0y0, then

dG.p; Œxz[Œyz/

3 2: These inequalities giveı.T /3=2and, hence,

ı C1.1; 2; : : : ; k/

3

2:

Since we have proved the converse inequality, we conclude that the equality holds.

As usual, thecomplementGof the graphGis defined as the graph withV G

DV .G/and such thate2E G

if and only ife…E.G/. We are going to bound the hyperbolicity constant of the complement of every infinite circulant graph. In order to do it, we need some preliminaries.

For any graphG, we define,

diamV .G/WDsup˚dG.v; w/jv; w2V .G/ ; diamGWDsup˚dG.x; y/jx; y2G : We need the following well-known result (see a proof,e.g., in [36, Theorem 8]).

Theorem 3.9. In any graphGthe inequalityı.G/.diamG/=2holds.

We have the following direct consequence.

Corollary 3.10. In any graphGthe inequalityı.G/.diamV .G/C1/=2holds.

From [34, Proposition 5 and Theorem 7] we deduce the following result.

Lemma 3.11. LetGbe any graph with a cycleg. IfL.g/3, thenı.G/3=4. IfL.g/4, thenı.G/1.

We say that a vertexvof a graphGis acut-vertexifGn fvgis not connected. A graph istwo-connectedif it does not contain cut-vertices.

We need the following result in [26, Proposition 4.5 and Theorem 4.14].

Theorem 3.12. Assume thatGis a two-connected graph. ThenGverifiesı.G/ D 1if and only ifdiamG D 2. Furthermore,ı.G/1if and only ifdiamG2.

Definition 3.13. Let us consider integers k 1and 1 a1 < a2 < < ak. We say thatfa1; a2; : : : ; akg

is a1-modulatedsequence if for everyx; y 2 Zwithjyj … fa1; a2; : : : ; akgwe havejxj … fa1; a2; : : : ; akgor jx yj … fa1; a2; : : : ; akg.

We have the following sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph. In particular, they show that the complement of infinite circulant graphs are hyperbolic.

Theorem 3.14. For any integersk1and1a1< a2< < akwe have

1ı C1.a1; a2; : : : ; ak/

3

2:

Furthermore,ı C1.a1; a2; : : : ; ak/

D1if and only iffa1; a2; : : : ; akgis1-modulated. If there is1j < ak=5

withj; 5j … fa1; a2; : : : ; akgand2j; 3j; 4j 2 fa1; a2; : : : ; akg, thenı C1.a1; a2; : : : ; ak/

D3=2.

Proof. DefineG WDC1.a1; a2; : : : ; ak/. Givenu; v 2 Z;considerw 2 Zwithw > uCakandw > vCak. SinceŒu; w; Œv; w …E.G/, we haveŒu; w; Œv; w2E G

anddG.u; v/dG.u; w/CdG.w; v/D2. Hence, diamV G

2and Corollary 3.10 givesı G

(13)

SinceŒ0; akC1; ŒakC1; 2akC2; Œ2akC2; 3akC3; Œ3akC3; 0…E.G/, we haveŒ0; akC1; ŒakC

1; 2akC2; Œ2akC2; 3akC3; Œ3akC3; 02E G

. Since the cycleC WD f0; akC1; 2akC2; 3akC3; 0gin

Ghas length4, Lemma 3.11 givesı G

1.

SinceGis a circulant graph, the sequencefa1; a2; : : : ; akgis1-modulated if and only if for everyx; y1; y22Z withjy2 y1j … fa1; a2; : : : ; akg, we havejx y1j … fa1; a2; : : : ; akgorjx y2j … fa1; a2; : : : ; akg. This happens if and only ifdG.x; Œy1; y2/ 1for everyx2V G

andŒy1; y22E G

, and this condition is equivalent to diamV G

2. SinceGis a two-connected graph, Theorem 3.12 that diamG2if and only ifı G

1. Since ı G

1, we conclude thatı G

D1if and only iffa1; a2; : : : ; akgis1-modulated.

Assume that there is1 j < ak=5with j; 5j … fa1; a2; : : : ; akgand2j; 3j; 4j 2 fa1; a2; : : : ; akg, and consider the cycleC WD f0; j; 2j; 3j; 4j; 5j; 0ginGwith length6. Letxandybe the midpoints of the edgesŒ2j; 3j  andŒ5j; 0, respectively. SincedG.Œ2j; 3j ; Œ5j; 0/D2,dG.x; y/D3andCcontains two geodesicsg1; g2joining

xandy, withg1\V G

D f0; j; 2jgandg2\V G

D f3j; 4j; 5jg. SincedG.j;f3j; 4j; 5jg/ 2, we have ı G

dG.j; g2/DdG.j;fx; yg/D3=2, and we concludeı G

D3=2.

Since Paley graphs is an important class of circulant graph, which is attracting great interest in recent years (see,e.g., [45]), we finish this paper with a result on the hyperbolicity constant of Paley graphs.

Recall that the Paley graph of orderqwithqa prime power is a graph onqnodes, where two nodes are adjacent if their difference is a square in the finite fieldGF .q/. This graph is circulant whenq1.mod4/. Paley graphs are self-complementary, strongly regular, conference graphs, and Hamiltonian.

Proposition 3.15. For any Paley graphGwe have

1ı.G/ 3

2:

Proof. Let us denote bynthe cardinality ofV .G/.

Since Paley graphs are self-complementary (the complement of any Paley graph is isomorphic to it), the degree of any vertex is.n 1/=2. Hence, givenu; v 2 V .G/withŒu; v… E.G/, there exists a vertexw 2 V .G/with dG.u; w/DdG.u; v/D1, and we conclude that diamV .G/2. Therefore, Corollary 3.10 givesı.G/3=2.

SinceGis a Hamiltonian graph, there exists a Hamiltonian cycleg. SinceL.g/ Dn5, Lemma 3.11 gives ı.G/1.

Acknowledgement: The authors thank the referees for their deep revision of the manuscript. Their comments and suggestions have contributed to improve substantially the presentation of this work. This work is supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

The first author is supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

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