On the hyperbolicity constant in graphs

If X is a geodesic metric space and 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 the Gromov sense) if any side of T is contained in a δneighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp 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 edge number, diameter and cycles. As a consequence of the study, we show that if G is any graph with m edges with lengths {lk}m k=1, then δ(G) ≤ m k=1 lk/4, and δ(G) = m k=1 lk/4 if and only if G is isomorphic to Cm. Moreover, we prove the inequality δ(G) ≤ 1 2 diam G for every graph, ...

• Type: Journal
• Year: 2011
• Where: DM
• Authors: José M. Rodríguez, Jose Maria Sigarreta, Jean-Marie Vilaire, María Villeta
• Keywords: DM 2011, Education, Geodesic, Geodesic Triangle, Precise Value