Mathematical Properties of the Hyperbolicity of Circulant Networks

Abstract EN

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.

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.