## Publication: The hyperbolicity constant of infinite circulant graphs

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##### Publication date

2017-06-09

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Walter de Gruyter GmbH

##### Abstract

If X is a geodesic metric space and x(1), x(2), x(3) is an element of 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 delta-hyperbolic (in the Gromov sense) if any side of T is contained in a delta-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.

##### Description

##### Keywords

Geodesics, Circulant graph, Gromov hyperbolicity, Infinite graphs

##### Bibliographic citation

Rodrı́guez, J. M., & Sigarreta, J. M. (2017). The hyperbolicity constant of infinite circulant graphs. Open Mathematics, vol. 15, no. 1, 2017, pp. 800-814