We present a study of 16 planetary nebulae (PNe) where fullerenes have been detected in their Spitzer Space Telescope spectra. Among the 16 PNe studied, we present the first detection of C sub(60) (and possibly also C sub(70)) fullerenes in the PN M 1-60 as well as of the unusual ~6.6, 9.8, and 20 mum features (attributed to possible planar C sub(24)) in the PN K 3-54. We do not find a metallicity dependence on the estimated fullerene abundances. The observed C sub(60) intensity ratios in the Galactic sources confirm our previous finding in the MCs that the fullerene emission is not excited by the UV radiation from the central star. With the data at hand, we suggest that the most likely explanation for the formation of fullerenes and graphene precursors in PNe is that these molecular species are built from the photochemical processing of a carbonaceous compound with a mixture of aromatic and aliphatic structures similar to that of hydrogenated amorphous carbon dust.