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 δ-neighborhood 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 } . The main result of this paper is the inequality δ(G) ≤ δ(L(G)) for the line graph L(G) of every graph G. We prove also the upper bound δ(L(G)) ≤ 5δ(G) + 3lmax, where lmax is the supremum of the lengths of the edges of G. Furthermore, if every edge of G has length k, we obtain δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 5k/2.