Collinear Néel-type ordering in partially frustrated lattices

We consider two partially frustrated $S=$ $\frac{1}{2}$ antiferromagnetic spin systems on the triangular and pentagonal lattices. In an elementary plaquette of the two lattices, one bond has exchange interaction strength α $(\alpha <~1)$ whereas all other bonds have exchange interaction strength unity. We show that for α less than a critical value ${\alpha }_c,$ collinear Néel-type ordering is possible in the ground state. The ground-state energy and the excitation spectrum have been determined using linear spin-wave theory based on the Holstein-Primakoff transformation.