Hyperbolicity of Direct Products of Graphs

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

The hyperbolicity constant of infinite circulant graphs

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.

Mathematical Properties of the Hyperbolicity of Circulant Networks

IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. 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.