  We consider two partially frustrated S = 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 $\alpha$ ($\alpha \leq
1$) whereas all other bonds have exchange interaction strength unity. We show
that for $\alpha$ less than a critical value $\alpha_{c}$, collinear
N\'eel-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.
