Mathematical Properties of the Hyperbolicity of Circulant Networks

Abstract

If is a geodesic metric space and , a geodesic triangle   is the union of the three geodesics , , and in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network 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 obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

1. Introduction

The first works on Gromov hyperbolic spaces deal with finitely generated groups (see [1]). Initially, Gromov spaces were applied to the study of automatic groups in the science of computation (see, e.g., [2]); indeed, hyperbolic groups are strongly geodesically automatic; that is, there is an automatic structure on the group [3]. Besides, hierarchical networks have been found to have “hidden hyperbolic structure” [4]. For a study of other parameters in complex networks, see [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 an Euclidean space of comparable dimension; the same holds for many complex networks; see [7]. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers (see [8]). Another important application of these spaces is the study of the spread of viruses on the Internet (see [9]). Furthermore, hyperbolic spaces are useful in secure transmission of information on the network (see [9]). The study of Gromov hyperbolic networks is a subject of increasing interest (see, e.g., [7, 9–20] and the references therein).

Hyperbolic spaces play an important role in geometric group theory and in the geometry of negatively curved spaces (see [1, 21]). The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 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, 21]).

If is a continuous curve in a metric space , the length of is defined as We say that is a geodesic if we have for every (then is equipped with an arc-length parametrization). The metric space is said to be geodesic if for every couple of points in there exists a geodesic joining them; we denote by any geodesic joining and ; 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 space is a network, then the edge joining the vertices and will be denoted by .

Along the paper we just consider graphs with every edge of length . In order to consider a network as a geodesic metric space, identify (by an isometry) any edge with the interval in the real line; then the edge (considered as a graph with just one edge) is isometric to the interval . Thus, the points in are the vertices and, also, the points in the interior of any edge of . In this way, any connected network has a natural distance defined on its points, induced by taking the shortest paths in , and we can see as a metric graph. If are in different connected components of , we define . Throughout this paper, denotes a simple (without loops and multiple edges) graph (not necessarily connected) such that every edge has length and . These properties guarantee that any connected component of any network is a geodesic metric space. Note that to exclude multiple edges and loops is not an important loss of generality, since [13, Theorems and ] reduce the problem of computing the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs.

If is a geodesic metric space and , the union of three geodesics , , and is a geodesic triangle that will be denoted by and we will say that and are the vertices of ; it is usual to write also . We say that is -thin if any side of is contained in the -neighborhood of the union of the two other sides. We denote by the sharp thin constant of ; that is, The space is -hyperbolic or satisfies the Rips condition with constant if every geodesic triangle in is -thin. We denote by the sharp hyperbolicity constant of ; that is, . If we have a triangle with two identical vertices, we call it a bigon; note that since this is a special case of the definition, every geodesic bigon in a -hyperbolic space is -thin. We say that is hyperbolic if is -hyperbolic for some ; then is hyperbolic if and only if If has connected components , then we define , and we say that is hyperbolic if .

In the classical references on this subject (see, e.g., [1, 21]) several different definitions of Gromov hyperbolicity appear, which are equivalent in the sense that if is -hyperbolic with respect to one definition, then it is -hyperbolic with respect to another definition (for some related to ). The definition that we have chosen has a deep geometric meaning (see, e.g., [1, 21]).

Trivially, any bounded metric space is -hyperbolic. A normed linear space is hyperbolic if and only if it has dimension one. A geodesic space is -hyperbolic if and only if it is a metric tree. If a complete Riemannian manifold is simply connected and its sectional curvatures satisfy for some negative constant , then it is hyperbolic. See the classical reference [1, 21] in order to find further results.

A network is circulant if 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 [22]. Every circulant graph is a vertex transitive graph and a Cayley graph [23].

The circulant is a natural generalization of the double loop network and was first considered by Wong and Coppersmith [24]. Our main interest in circulant graphs lies in 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 [25, 26]. 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 [27]. Theoretical properties of circulant graphs have been studied extensively and surveyed [25].

The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds of this constant for particular classes of graphs. For a general graph or a general geodesic metric space deciding whether or not a space is hyperbolic is usually very difficult. Therefore, it is interesting to relate the hyperbolicity with other classes of graphs. The papers [10, 14, 15, 20] prove, respectively, that chordal, -chordal, edge-chordal, and join graphs are hyperbolic. Moreover, in [10] it is shown that hyperbolic graphs are path-chordal graphs. The authors have proved in a previous work that every circulant graph is hyperbolic (and they obtain inequalities for the hyperbolicity constant of infinite circulant graphs). In this paper we obtain several sharp inequalities for the hyperbolicity constant of finite circulant networks; in some cases we characterize the graphs for which the equality is attained. Theorem 3 in Section 2 gives the precise value of the hyperbolicity constant of . Theorem 11 provides a sharp lower bound for and characterizes the graphs for which the equality is attained. It is well known that a graph is circulant if and only if its complement is circulant. Thus it is natural to study in this context the properties of general complement graphs. In Theorems 15 and 24 this kind of results appears and they are applied to circulant graphs in Corollary 25. We collect in Section 3 several sharp inequalities for the hyperbolicity constant of a large class of circulant graphs. In Theorem 28 good lower and upper bounds for appear, which are improved in Theorems 29, 30, and 31 with additional hypothesis. Furthermore, we obtain the precise value of the hyperbolicity constant of many circulant networks (see Theorems 3, 11, and 29 and Corollary 25).

2. Bounds for the Hyperbolicity Constant

Given any natural number , let be a set of integers such that , where denotes the lower integer part of .

We define the circulant network as the finite graph with vertices (or ) such that is the set of neighbors of each vertex . If , then is a regular graph of degree . For even, we allow ; in this case, is regular of degree .

The following result is well known (see, e.g., [28, Theorem ]).

Theorem 1. The circulant graph is connected if and only if .

If a circulant graph has connected components , then and are isomorphic for every , is also a circulant graph, and for every . Thus the condition connected is not a real restriction (unless if we deal with the complement graph of , as in Theorems 15 and 24 and Corollary 25).

As usual, by cycle we mean a simple closed curve, that is, a path with different vertices, unless the last one, which is equal to the first vertex.

We also need the following result in [18, Theorem ].

Theorem 2. If is the cycle graph with vertices, then .

The next result provides the precise value of for every value of and .

Theorem 3. If with , then the circulant graph has connected components, for every and if .
If , then has connected components and .

Proof. If , then it is clear that has connected components. If , then is the disjoint union of edges and . If , then is the disjoint union of graphs isomorphic to . Thus Theorem 2 gives .
If , then is the disjoint union of graphs isomorphic to , and Theorem 2 gives .

From [17, Proposition and Theorem ] we deduce the following result.

Lemma 4. Let be any graph with a cycle . If , then . If , then .

For the sake of completeness, we are going to give an idea of the proof of this lemma. We need a definition and a lemma. We say that a subgraph of is isometric if for every . It is easy to check that a subgraph of is isometric if and only if for every . Isometric subgraphs are very important in the study of hyperbolic graphs, as the following result shows.

Lemma 5 (see [18, Lemma ]). If is an isometric subgraph of , then .

Let us start with the idea of the proof of Lemma 4. If , then is an isometric subgraph, and Lemma 5 and Theorem 2 give . If , then the graph induced by is an isometric subgraph; thus, is isomorphic to either , , or without an edge, and Lemma 5 gives . Assume now that and there is no cycle in of length . Let be a curve with One can prove that is an isometric subgraph and Lemma 5 and Theorem 2 give .

By we mean that we have either or .

Definition 6. Given and one says that the sequence is -full if for every and there exists such that or .

For any graph , we define, as usual,

Definition 7. One says that a vertex of a graph is a cut-vertex if is not connected. A graph is two-connected if it does not contain cut-vertices. Given any edge in , let one consider the maximal two-connected subgraph containing it. One calls to the set of these maximal two-connected subgraphs the canonical T-decomposition of . One defines the effective diameter of as

Note that if is a two-connected graph, then and .

We need the following result in [12, Proposition and Theorem ].

Theorem 8. A graph verifies if and only if . Furthermore, if and only if .

We need the following result in [18, Theorem ].

Theorem 9. In any graph the inequality holds.

We have the following direct consequence.

Corollary 10. In any graph the inequality holds.

Denote by the set of vertices and midpoints of edges in .

Since Theorem 3 gives the precise value of , in order to study we just need to deal with the case .

We prove now a sharp lower bound for the hyperbolicity constant and we characterize the graphs for which this lower bound is attained.

Theorem 11. For any integers and such that is connected, one has and if and only if is -full.

Proof. We are going to prove that contains a cycle with length at least .
Assume first that for some . Thus, contains a cycle with length at least .
Assume now that for every . Seeking for a contradiction assume that for some . Then and , contradicting . So, for every . If for some , then . If for some , then . Since , we deduce , , and , and there exists a positive integer such that , , and . Since is connected, Theorem 1 gives that . Hence, , and the cycle with consecutive vertices has length .
Thus, contains a cycle with length at least in any case, and Lemma 4 gives .
Denote by the vertices of .
Assume first that is -full. We are going to show that for every and . Since is a circulant graph, we can assume by symmetry. Since , there exist and such that belongs to . Thus there exists such that or , and we have . Hence, . Therefore, for every and , and we conclude for every and for every . Thus and Theorem 9 gives , and we conclude .
Assume now that . Since is a two-connected graph, Theorem 8 gives . Hence, for every and . Consider and and let be the midpoint of . Therefore, implies . Thus there exists such that or , and we conclude that is -full.

In [11, Theorem ] the following result appears.

Theorem 12. For every hyperbolic graph , is a multiple of .

Theorems 11 and 12 have the following consequences.

Corollary 13. For any integers and such that is connected, one has if and only if is not -full.

Corollary 14. For any integers , , and , one has and .

Proof. If we have either or , then and , and so and Corollary 10 gives . Theorem 11 gives the converse inequality.
Assume that . If , then , and the previous argument gives . If , then we have either or . If , then we have , , , , , , and , and Theorem 11 gives . If , then we have , , , , , , , and , and Theorem 11 gives .
Finally, consider , , , and . Since , , , and , Corollary 13 gives . One can check that , and Theorem 9 gives .

It is well known that a graph is circulant if and only if its complement is circulant. Thus it is natural to study in this context the properties of general complement graphs. In Theorems 15 and 24 this kind of results appears and they are applied to circulant graphs in Corollary 25.

As usual, the complement of the (connected or nonconnected) graph is defined as the graph with such that if and only if .

Theorem 15. If is a graph with , then is connected and , and this inequality is sharp.

Proof. Seeking for a contradiction assume that there exists an edge such that for every . Choose with . Thus , which is a contradiction. Hence, for each edge there exists with .
Fix . If , then , , and . If , then there exists with . Thus , , and . Hence, is connected, , and Corollary 10 gives the inequality.
The following family of graphs shows that this inequality is sharp. Let be an even integer and . Since is a -regular graph and , the Moore’s bound gives . Hence, we have proved that and it suffices to show that . Denote by the vertices of and consider the cycle in with length and consecutive vertices . Let , be the midpoints of and , respectively, and , the two geodesics contained in joining and with vertices and , respectively. Since , we have and

Theorem 24 below gives more information than Theorem 15 for nonconnected graphs. In order to prove it, we need some technical results.

Lemma 16. If is a nonconnected graph with connected components and for every , , and , then .

Proof. Note that it suffices to check that for every and .
Let be the midpoint of for some . If , then since . If for some , then .
Let be the midpoint of with , , and . If , then . If , then .
Hence, for every and , and we conclude .

Note that a connected graph satisfies for every , if and only if . A nonconnected graph satisfies this property if and only if .

We have the following direct consequence of Lemma 16.

Corollary 17. If is a nonconnected graph with connected components and for every , then .

Let and be two graphs with . We recall that the graph join of and is the graph such that and two different vertices and of are adjacent if and , or or .

The argument in the proof of Lemma 16 gives the following result.

Corollary 18. If and are graphs with and , then .

Definition 19. One says that a nonconnected graph with connected components satisfies the -vertex-edge property if we have either , for every , , and , or , , and , for every , where are the connected components of , or , , and for every where are the connected components of .

Theorem 20. Let be a nonconnected graph with connected components . Then if and only if satisfies the -vertex-edge property.

Proof. Assume that for every , , and . Lemma 16 gives .
Assume now that , , and for every where are the connected components of . If , then we define for each the graph . Corollary 18 gives that . Since is the canonical -decomposition of , we conclude that .
If , , and for every where are the connected components of , then the previous argument also gives .
Finally, assume that .
Note that has a cut-vertex if and only if and we have either and being nonconnected or and being nonconnected.
Assume that has a cut-vertex. By symmetry we can assume that , , and is nonconnected. Let be the connected components of , and consider defined as before. Thus for every . Seeking for a contradiction assume that for some . Then for some and , and we conclude , which is a contradiction. Hence, for every , and satisfies the -vertex-edge property.
Assume now that does not have cut-vertices. Thus . Seeking for a contradiction assume that for some , , and . If is the midpoint of , then , and we conclude , which is a contradiction. Hence, for every , , and , and satisfies the -vertex-edge property.

Consider the set of geodesic triangles in that are cycles such that the three vertices of the triangle belong to .

The following result appears in [11, Theorem ].

Theorem 21. For any hyperbolic graph there exists a geodesic triangle such that .

The following result in [17, Theorem ] will be useful.

Theorem 22. If is a graph with , then one has either or . Furthermore,(i) if and only if is a tree;(ii) if and only if and every cycle in has length .

Definition 23. Given a graph with , one says that a subgraph contains a maximal triangle and there exists a geodesic triangle in that is a cycle such that , , and is contained in .

Note that if contains a maximal triangle , then we can rename the vertices of in order to guarantee that there exists such that , , and . Furthermore, and .

The following result provides the precise value of for every nonconnected graph .

Theorem 24. If is a nonconnected graph with connected components , then is connected and . Furthermore,(i) if and only if and and are complete graphs and we have or ;(ii) if and only if and we have either , , or , , and being isomorphic to a complete graph without a nonempty set of pairwise disjoint edges, or , , and being isomorphic to a complete graph without a nonempty set of pairwise disjoint edges;(iii) if and only if and satisfies the -vertex-edge property;(iv) if and only if and does not contain a maximal triangle for every ;(v) if and only if contains a maximal triangle for some .

Proof. Theorem 15 gives that is connected and . Furthermore, the argument in the proof of Theorem 15 provides that and thus .
is a tree if and only if and and are complete graphs and we have or . This gives the first item.
is not a tree and every cycle in has length if and only if we have either , (if does not have cut-vertices), or , , and being isomorphic to a complete graph without a nonempty set of pairwise disjoint edges, or , , and being isomorphic to a complete graph without a nonempty set of pairwise disjoint edges (if has a cut-vertex). Thus Theorem 22 gives the second item.
Assume that . By Theorem 9, we have and . By Theorem 21, there exist a geodesic triangle that is a cycle in and such that and . Since and , we have and is the midpoint of . Thus and . Besides, and .
Let be the midpoint of with , , and . If , then . If , then . Hence, for every and for every .
Let be the midpoint of for some and for some . Thus, and for every .
Therefore, there exist and such that and are the midpoints of and , respectively. By symmetry, we can assume that , and so . Since is a cycle, . If , then , which is a contradiction. Hence, . If there exists a vertex with , then , which is a contradiction. Therefore, and so is a maximal triangle in .
Assume now that contains a maximal triangle for some . Thus and, since , we conclude .
Theorem 20 gives that if and only if satisfies the -vertex-edge property. By Theorem 8, if and only if satisfies the -vertex-edge property. Theorem 12 gives if and only if . Hence, if and only if and satisfies the -vertex-edge property.
Finally, the previous results and Theorem 12 provide the characterization of the graphs with .

Theorem 24 has the following consequence for circulant graphs.

Corollary 25. Fix integers and such that is nonconnected, and consider integers and such that has connected components isomorphic to . Then is a connected circulant graph and . Furthermore,(i) if and only if we have either or being a complete graph;(ii) if and only if and does not contain a maximal triangle;(iii) if and only if contains a maximal triangle.

Proof. Since and . Hence, satisfies the -vertex-edge property if and only if we have either or being the complete graph with vertices. Thus Theorem 24 gives the result.

3. Bounds for the Hyperbolicity Constant If

The following result is well known (see, e.g., [28, Proposition ]).

Theorem 26. If is such that for some with , then there exists a circulant graph isomorphic to with .

Hence, it is natural to find bounds for the hyperbolicity constant of . We will need the following result.

If is a subgraph of and , we denote by the degree of the vertex in the subgraph induced by .

Theorem 27 (see [12, Theorem ]). Let be any graph. Then if and only if there exist a cycle in with length and a vertex such that .

The following result provides good lower and upper bounds if .

Theorem 28. For any integers and one hasif , andotherwise. The second equality in (9) is attained if , is odd, and . The second equality in (10) is attained if and is an odd multiple of .

Proof. Let us denote by the vertices of , and let us denote by the subgraph of with and .
We prove first the upper bounds.
We are going to find an upper bound of . We want to remark that it is not possible to find a simple formula for (and not even for , see [28]).
Fix a vertex , and denote by the vertices with (if is a multiple of , then ); therefore, and . For each real number with , define as the point in with and .
Assume that (the case is trivial).
We have If , then we define as the vertex verifying and . Assume that . Then and we have and ; hence, for , Using a symmetric argument we obtain the same inequality for .
Hence, one can check that holds for every , if or . Since is equal to the closed ball or radius and center for every , we conclude in this case.
Assume now that . Then and we have and ; hence, for , Using a symmetric argument we obtain the same inequality for . Hence, holds for every , and we conclude in this case.
If ( is an odd multiple of ), then a similar argument gives for every . If is the midpoint of , then the previous argument gives Hence, we also obtain in this case.
If , then a similar argument gives for every . If is the midpoint of , then the previous argument gives Thus, we obtain in this case.
Therefore, Theorem 9 gives the desired inequalities.
Assume now that and is an odd multiple of . Define and . Fix a vertex , and denote by vertices with for ,   for ,   for ,   for ,   for ,   for , and . Define Since is an odd multiple of , we have Hence, if is even and if is odd; let be the midpoint of and define and . Then and are geodesics and . Let be the geodesic bigon and the midpoint of . We have Since we have proved the converse inequality, we conclude that the equality holds.
Assume that , is odd and . We obtain the equality by using a similar bigon to the previous case, with .
Finally, we prove the lower bound. By Theorem 11, we can assume that .
Let us define , , and (if is a multiple of , then ). Consider the geodesics We define an appropriate geodesic in the following way. A geodesic can be obtained aswith and (if , then the part of with negative numbers does not appear).
Let us define the finite sequence in the following way: if , and otherwise (i.e., we define ). It is clear that and .
Consider . If , then If , then If , then and so, Let us definefor (if , then is a single edge). We have Since joins and , and , we consider the geodesic .
Since we have for every .
Since , we have and we conclude that for every . Let be the midpoint of . Since for every ,

The lower bound in Theorem 28 can be improved for .

Theorem 29. For any integers with one hasThe equality in (33) is attained if .

Proof. Let us denote by the vertices of , and let us denote by the subgraph of with and .
Let , , and . Thus (if is a multiple of , then ); therefore, and . Note .
If , then Theorem 28 gives (33).
Assume that . Define as the point in with . One can check that . Define as the geodesic in with and the geodesic in with . Let and be geodesics in contained in . Thus and are two geodesics in joining and . If is the midpoint of , then Theorem 28 gives that the equality in (33) is attained if .

The upper bounds in Theorem 28 can be improved for large enough.

Theorem 30. Consider any integers and . Assume that if and if . Then one has

Proof. By Theorem 11, it suffices to prove the upper bound. Recall that if , then is a regular graph with degree and that if , then is regular with degree . In any case we have for every . Therefore, given any with , there exists a vertex with ; thus and we conclude that . Then Corollary 10 gives .

Note that by Theorem 12, under the hypothesis in Theorem 30, the possible values for are just , , and .

Finally, the next result estimates the hyperbolicity constant if .

Theorem 31. For any integers one has if , and if .

Proof. Let us denote by the vertices of .
It is easy to check that and Corollary 10 gives the upper bounds.
In order to prove the lower bounds, assume first that is even (and then ).
Define , as the geodesics in joining and with and . If is the midpoint of , then Assume now that is odd (and then ) and let be the midpoint of the edge .
Define , as the geodesics in joining and with and . If is the midpoint of , then

4. Conclusions

In this paper we study the hyperbolicity constant of an important class of networks: circulant graphs. We obtain several sharp inequalities for the hyperbolicity constant and in some cases we characterize the graphs for which the equality is attained.

Theorem 3 in Section 2 gives the precise value of the hyperbolicity constant of . Theorem 11 provides a sharp lower bound for and characterizes the graphs for which the equality is attained. It is well known that a network is circulant if and only if its complement is circulant. Thus it is natural to study in this context the properties of general complement graphs. In Theorems 15 and 24 this kind of results for general networks appears and they are applied to circulant graphs in Corollary 25.

We collect in Section 3 several sharp inequalities for the hyperbolicity constant of a large class of circulant graphs. In Theorem 28 good lower and upper bounds for appear, which are improved in Theorems 29, 30, and 31 with additional hypothesis. Furthermore, we obtain the precise value of the hyperbolicity constant of many circulant networks (see Theorems 3, 11, and 29 and Corollary 25).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was supported in part by a grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), Mexico.