Polybenzene was described by O'Keeffe et al., as an embedding of a 6.82 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 C60, the reference structure in Nanoscience. They also described a 6.82 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 C60 but also to diamonds (the classical diamond D6 and the pentagon-based diamond D5), 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.