On the hyperbolicity constant in graphs

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, for every geodesic triangle T in X , every side of T is contained in a δ -neighborhood of the union of the other two sides. We denote by δ ( X ) the sharpest hyperbolicity constant of X , i.e. δ ( X ) ≔ inf { δ ≥ 0 : X  is  δ -hyperbolic } . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths { l k } k = 1 m , then δ ( G ) ≤ ∑ k = 1 m l k / 4 , and δ ( G ) = ∑ k = 1 m l k / 4 if and only if G is isomorphic to C m . Moreover, we prove the inequality δ ( G ) ≤ 1 2 diam G for every graph, and we use this inequality in order to compute the precise value δ ( G ) for some common graphs.