  Exact construction of one electron eigenstates with flat, non-dispersive
bands, and localized over clusters of various sizes is reported for a class of
quasi-one dimensional looped networks. Quasiperiodic Fibonacci and Berker
fractal geometries are embedded in the arms of the loop threaded by a uniform
magnetic flux. We work out an analytical scheme to unravel the localized single
particle states pinned at various atomic sites or over clusters of them. The
magnetic field is varied to control, in a subtle way, the extent of
localization and the location of the flat band states in energy space. In
addition to this we show that, an appropriate tuning of the field can lead to a
re-entrant behavior of the effective mass of the electron in a band, with a
periodic flip in its sign.
