  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.
