Scaled Gromov hyperbolic graphs

Jonckheere, Edmond; Lohsoonthorn, Poonsuk; Bonahon, Francis

doi:10.1002/jgt.20275

Cited by 70 publications

(73 citation statements)

References 14 publications

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“…7 }, {v (7) 4 , v 6 }, {v (7) 4 , v Let C 9 be the set of graphs obtained from C 9 by adding a (proper or not) subset of the set of edges {v (9) 2 , v (9) 6 }, {v (9) 2 , v (9) 9 }, {v (9) 4 , v (9) 6 }, {v (9) 4 , v (9) 9 } . Define: F 9 :={graphs containing, as the induced subgraph, an isomorphic graph to some element of C 9 }.…”

Section: The Case Of 5/4mentioning

confidence: 99%

“…In this way, a simple concept led to a very rich general theory (see [1][2][3][4]) and, in particular, made hyperbolic spaces applicable to graphs. The theory has also been extensively used in discrete spaces like trees, the Cayley graphs of many finitely-generated groups and random graphs (see, e.g., [6][7][8][9]). …”

Section: Introductionmentioning

confidence: 99%

“…(see, for example, [6][7][8][9]14,[18][19][20][21][25][26][27][28][29][30][31][32][33][34][35]). x, y ∈ J(G) \ V(G), z ∈ J(G), …”

Section: Introductionmentioning

confidence: 99%

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Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G M is hyperbolic and that δ(G M ) is comparable to diam(G M ). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 ≤ δ(G M ) ≤ 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on δ(G) just in terms of δ(G M ) is obtained.

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Section: The Case Of 5/4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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“…Moreover, the authors in Kennedy et al 17 (using the slim triangles condition for hyperbolicity) and in Jonckheere et al 56 (using the four-point condition) argue that the hyperbolicity of the graph is ''actually'' present when the value of is much smaller than the graph's diameter. They specify that for a graph to be hyperbolic the value of /diam(G) must asymptotically scale to zero.…”

Section: Lemmamentioning

confidence: 99%

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

Alrasheed

Dragan

2016

Journal of Algorithms & Computational Technology

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Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov's notion of -hyperbolicity but also by analyzing its relationship to other graph's parameters. This new perspective allows us to classify graphs with respect to their hyperbolicity and to show that many biological networks are hyperbolic. Then we introduce the eccentricity-based bending property which we exploit to identify the core vertices of a graph by proposing two models: the maximum-peak model and the minimum cover set model. In this extended version of the paper, we include some new theorems, as well as proofs of the theorems proposed in the conference paper. Also, we present the algorithms we used for each of the proposed core identification models, and we provide more analysis, explanations, and examples.

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“…Intuitively, a Gromov hyperbolic graph is a graph that has the property that it looks like a tree when viewed at a distance, where the concept of "viewing at a distance" is formalized in large-scale geometry, also referred to as coarse geometry or asymptotic metric theory [15]. The importance of Gromov hyperbolic graphs is that scale free, and hence Internet, graphs have that property [18], [23], [19]. Since Gromov hyperbolic graphs are a generalization of trees, the natural question is, "What is the effective resistance of a Gromov hyperbolic graph?"…”

mentioning

confidence: 99%

Effective resistance of Gromov-hyperbolic graphs: Application to asymptotic sensor network problems

Jonckheere

Lou

Hespanha

et al. 2007

2007 46th IEEE Conference on Decision and Control

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Abstract-The technique of effective resistance has seen growing popularity in problems ranging from escape probability of random walks on graphs to asymptotic space localization in sensor networks. The results obtained thus far deal with such problems on Euclidean lattices, on which their asymptotic nature already reveals that the crucial issue is the large scale behavior of such lattices. Here we investigate how such results have to be amended on a class of graphs, referred to as Gromov hyperbolic, which behave in the large scale as negatively curved Riemannian manifolds. It is argued that Gromov hyperbolic graphs occur quite naturally in many situations. Among the results developed here, we will mention the nonvanishing probability of escape of a random walk to a Cantor set Gromov boundary and the facts that the space localization error of sensors networked in a Gromov hyperbolic fashion grows linearly with the distance to a sensor whose geographical position is known, but would become uniformly bounded in an idealized situation in which the geographical locations of the nodes at the Gromov boundary are known

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Scaled Gromov hyperbolic graphs

Cited by 70 publications

References 14 publications

Gromov Hyperbolicity in Mycielskian Graphs

Gromov Hyperbolicity in Mycielskian Graphs

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

Effective resistance of Gromov-hyperbolic graphs: Application to asymptotic sensor network problems

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