If 𝑋 is a geodesic metric space and $x_1,x_2,x_3 \in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ 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 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{$\delta\geq 0$ : $X$ is $\delta$-hyperbolic}. In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.