Recent Results on Hyperbolicity on Unitary Operators on Graphs

Abstract

For a geodesic metric space X and for x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { 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 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