Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

AbstractIn this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface

Gromov hyperbolicity in strong product graphs

If X is a geodesic metric space and x1; x2; x3 2 X, a geodesic triangle T = fx1; x2; x3g is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is -hyperbolic (in the Gromov sense) if any side of T is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by (X) the sharp hyperbolicity constant of X, i.e. (X) = inff > 0 : X is -hyperbolic g : In this paper we characterize the strong product of two graphs G1 G2 which are hyperbolic, in terms of G1 and G2: the strong product graph G1 G2 is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between (G1 G2), (G1), (G2) and the diameters of G1 and G2 (and we nd families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs

Gromov hyperbolicity in directed graphs

In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov hyperbolicity and the geodesic stability. This theorem has potential applications to the development of solutions for secure data transfer on the internetSupported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spai