By means of first-principles electronic structure calculations, the ordered graphene nanomeshes with patterned hexagonal vacancy holes are theoretically studied to explore the modification mechanism of electrical conduction on graphene atomic monolayers. According to pseudopotential plane wave first-principles scheme based on density functional theory, the band structures of graphene nanomeshes are calculated to analyze the electrical conductance in correlation with the superlattice symmetry and vacancy hole magnetism. Based on the structural features and topological magnetism of Y-shaped nodes between the nanopores on the atomic monolayer of graphene, the graphene nanomeshes are classified into three types. The quadruplet degeneracy and splitting of electronic states at Brillouin zone center are investigated by comparing the band structures of graphene nanomeshes and analogical superlattices. The effects of inversion symmetry and supercell size on the opening band-gap at Dirac cone are elaborately analyzed with the consideration of antiferromagnetic coupling and hydrogen passivation at the magnetic edge of nanopores on graphene nanomeshes. The band-structure calculation results indicate that the (3m, 3n) (m and n are integers) superlattices have fourfold degenerate electronic states at center point of Brillouin zone, which can be effectively splitted by regularly arranging porous atomic vacancy to make the (3m, 3n) nanomesh, resulting in adjustable band-gap no matter whether or not the sublattices keeping in equivalence. In the nanomeshes formed by patterned holes with magnetic edge, the antiferromagnetic coupling adds a quantum parameter to the inversion symmetry so as to break the sublattice equivalence, opening band-gap at the twofold degenerate K point. Nevertheless, the hydrogen passivation at the edge of magnetic nanopores will convert the magnetic graphene nanomeshes into non-magnetic and eliminate the band-gap at K point. The band-gap of graphene nanomeshes could also be controlled by changing the density of nanopores, suggesting a graphene nanomaterial with adjustable band-gap that can be designed by controlling the mesh pore spacing. The graphene nanomeshes represent a new mechanism of forming band-gap and thus promise a strategy for achieving special electrical properties of graphene nanostructures. These results also theoretically demonstrate that the nano-graphene is a prospective candidate with flexibly adjustable electrical properties for realizing multivariate applications in new-generation nano-electronics.