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Gromov hyperbolic cubic graphs

100%

Pestana D., Rodríguez J., Sigarreta J., Villeta M.

Open Mathematics

2012

tom 10

nr 3

1141-1151

If X is a geodesic metric space and x 1; x 2; x 3 ∈ X, a geodesic triangle T = {x 1; x 2; x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.

The hyperbolicity constant of infinite circulant graphs

100%

Rodríguez J. M., Sigarreta J. M.

Open Mathematics

2017

tom 15

nr 1

800-814

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. 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.

Ograniczanie wyników

2 Open Mathematics

1 Pestana D.

1 Rodríguez J.

1 Rodríguez J. M.

1 Sigarreta J.

1 Sigarreta J. M.

1 Villeta M.

1 2017

1 2012

Wyniki wyszukiwania

Gromov hyperbolic cubic graphs

The hyperbolicity constant of infinite circulant graphs