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Recent results on hyperbolicity on unitary operators on graphs

Abstract:

For a geodesic metric space X and for 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 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. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X):= sup{δ(T ): T is a geodesic triangle in X }. In this paper, we collect previous results and prove new theorems on the hyperbolic constant of some important unitary operators on graphs.