Systematic enumeration of crystalline networks with only sp2 configuration in cubic lattices

Systematic enumeration of crystalline networks with some special topological characters is of considerable interest in both mathematics and crystallography. Based on the restriction of lattice in cubic and inequivalent nodes not exceeding three, a simple method is proposed for systematic searching for three-dimensional crystalline networks with only sp2-configuration nodes (C-sp2-TDTCNs). We systematically scan the cubic space groups from No.195 to No.230 and find many C-sp2-TDTCNs including all the previously proposed cubic ones. These C-sp2-TDTCNs are topologically intriguing and can be considered as good templates for searching carbon crystals with novel properties, predicting high pressure phases of element nitrogen and designing three-dimensional hydrocarbon crystals. Structure optimizations are considered by regrading these C-sp2-TDTCNs as carbon crystals and the corresponding energetic stability of these carbon crystals are evaluated, using the the density functional theory (DFT) based first-principle calculations. Our results are of wide interests in mathematics, condensed physics, crystallography and material science.