This paper provides the first analysis of the differentially private computation of three centrality measures, namely eigenvector, Laplacian and closeness centralities, on arbitrary weighted graphs, using the smooth sensitivity approach. We do so by finding lower bounds on the amounts of noise that a randomised algorithm needs to add in order to make the output of each measure differentially private. Our results indicate that these computations are either infeasible, in the sense that there are large families of graphs for which smooth sensitivity is unbounded; or impractical, in the sense that even for the cases where smooth sensitivity is bounded, the required amounts of noise result in unacceptably large utility losses.