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 internet.

On the Gromov-Hausdorff limit of metric spaces

In this paper, we show that a class of metric spaces determined by a continuous function f, which defines on the metric space of all real, n × n-matrices m is closed under the Gromov-Hausdorff convergence. This conclusion can be used to prove some metric properties of metric space is stable under the Gromov-Hausdorff convergence. Secondly, we consider the stability problem in Gromov hyperbolic space and show that if a sequence of Gromov hyperbolic spaces (Xn, dn) is said to converge to (X, d) in the sense of Gromov-Hausdorff convergence, then the Gromov hyperbolicity δ(Xn) of (Xn, dn) tends to the Gromov hyperbolicity δ(X) of (X, d).