Hyperbolicity in the corona and join of graphs

If X is a geodesic metric space and x1,x2,x3∈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 delta-hyperbolic (in the Gromov sense) if any side of T is contained in a delta-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by delta(X) the sharp hyperbolicity constant of X, i.e. delta(X) = inf{delta ≥ 0: X is delta-hyperbolic}. In this paper we characterize the hyperbolic product graphs for graph join G + H and the corona G⊙H:G+H is always hyperbolic, and G⊙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⊙H.

Hyperbolicity in the corona and join of graphs Articles