Bounds on the Hyperbolicity Constant

Abstract If X is a geodesic metric space and 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 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. If X is hyperbolic, we denote by δ ( X ) the sharp hyperbolicity constant of X, i.e. δ ( X ) = inf ⁡ { δ ≥ 0 : X is δ -hyperbolic } . To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by G ( n , m ) the set of graphs G with n vertices and m edges, and such that every edge has length 1. In this work we estimate A ( n , m ) : = min ⁡ { δ ( G ) | G ∈ G ( n , m ) } and B ( n , m ) : = max ⁡ { δ ( G ) | G ∈ G ( n , m ) } . In particular, we obtain good bounds for A ( n , m ) and B ( n , m ) , and we compute the precise value of A ( n , m ) for many values of n and m. In addition, we obtain an upper bound of the size of any graph in terms of its diameter and its order.

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