Hyperbolicity in the corona and join of graphs.

If X is a geodesic metric space and $${x_1, x_2, x_3 \in X}$$ , a geodesic triangle T = { x, x, x} is the union of the three geodesics [ x x], [ x x] and [ x x] 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}. In this paper we characterize the hyperbolic product graphs for graph join G + H and the corona $${G\odot\mathcal H: G + H}$$ is always hyperbolic, and $${G\odot\mathcal H}$$ is hyperbolic if and only if G is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G + H and the corona $${G \odot \mathcal H}$$ . [ABSTRACT FROM AUTHOR]