By first-principles calculations, we investigated the electronic structures and magnetic properties of several tetravalent transition-metal monolayers with normal honeycomb, triangular-dodecagonal, and square-octagonal structures by considering the effects of spin-orbit coupling and electronic strong correlation of d orbitals. For both standard and corrected approaches, spin-polarized Dirac points contributed by d states appear in the monolayers with hexagonal lattice (honeycomb and 3-12 lattices), but for 4-8 lattices, Dirac points disappear, demonstrating that specific symmetries are required for forming Dirac cones. By adding the on-site Coulomb repulsion, the electronic correlation of d orbital is enhanced and thus the electronic localization increases, aggravating the spin splitting. For Hf3-12, the coexistence of massless Dirac fermions and massive heavy fermions is found. Moreover, the spin-orbit coupling destroys the degeneracy of two bands at K points, and the largest gap opening of 214 meV appears in Hf4-8 due to both Coulomb repulsion and spin-orbit coupling. Our results demonstrate that the spin splitting and gap opening depend on the lattice symmetry, bond length, electronic strong correlation, and spin-orbit coupling. These predicted structures provide new choices in synthesizing two-dimensional transition-metal materials, which has the potential applications in spintronic devices, quantum computation, hydrogen storage, and catalytic chemistry. (C) 2017 Elsevier B.V. All rights reserved.