Mathematical Properties of the Hyperbolicity of Circulant Networks

If X is a geodesic metric space and x(1), x(2), x(3) is an element of X, a geodesic triangle x = {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 x is contained in a delta-neighborhood of the union of the two other sides, for every geodesic triangle x 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.

Mathematical Properties of the Hyperbolicity of Circulant Networks Articles