  We investigate magnetic activity properties of 21 stars via medium resolution
optical spectra and long-term photometry. Applying synthetic spectrum fitting
method, we find that all targets are cool giant or sub-giant stars possessing
overall [M/H] abundances between $0$ and $-0.5$. We find that six of these
targets exhibit only linear trend in mean brightness while eight of them
clearly shows cyclic mean brightness variation. Remaining seven target appear
to exhibit cyclic mean brightness variation but this can not be confirmed due
to the long time scales of the predicted cycle compared to the current time
range of the photometric data. We further determine seasonal photometric
periods and compute average photometric period of each target. Analysed sample
in this study provides a quantitative representation of a positive linear
correlation between the inverse of the rotation period and the cycle period
normalized to the rotation period, on the log-log scale. We also observe no
correlation between the activity cycle length and the relative surface shear,
indicating that the activity cycle must be driven by a parameter rather than
the differential rotation. Our analyses show that the relative surface shear is
positively correlated with the rotation period and there is a noticeable
separation between main sequence stars and our sample. Compared to our sample,
the relative surface shear of a main sequence star is larger for a given
rotation period. However, dependence of the relative surface shear on the
rotation period appears stronger for our sample. Analysis of the current
photometric data indicates that the photometric properties of the observed
activity cycles in 8 targets seem dissimilar to the sunspot cycle.
