A graph operator is a mapping F : Gamma -> Gamma', where Gamma and Gamma' are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Lambda(G); subdivision graph, S (G); total graph, T (G); and the operators R (G) and Q (G). In particular, we get relationships such as delta(G) <= delta(R (G)) <= delta(G) + 1/2, delta(Lambda(G)) <= delta(Q (G)) <= delta(Lambda(G)) + 1/2, delta(S (G)) <= 2 d(R (G)) <= delta(S (G)) + 1 and delta(R (G)) - 1/2 <= d(L(G)) <= 5 delta(R (G)) + 5/2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.