Polybenzene was described by O'Keeffe et al., as an embedding of a 6.8(2) net in the infinite periodic minimal D-surface, with a single type of carbon atoms and was predicted to have a substantially lower energy per atom in comparison to C-60, the reference structure in Nanoscience. They also described a 6.8(2) net embedded in the periodic minimal P-surface. We give here a rational structure construction for three benzene-based units (a third one described here for the first time in literature) and the corresponding networks. Their stability, relative to C-60 but also to diamonds (the classical diamond D-6 and the pentagon-based diamond D-5), was calculated at the Hartree-Fock level of theory. The results confirmed the previous stability evaluation and support these structures for laboratory preparation. A Graph-theoretical description, in terms of Omega polynomial, of the three infinite networks is also presented.