  We have obtained and analyzed the electronic states, polarization function
and the plasmon excitations for $\alpha - \mathcal{T}_3$-based nanoribbons with
armchair termination. The calculated plasmon dispersions strongly depend on the
number of the atomic rows across the ribbon, and the presence of the energy gap
between the valence and conduction bands which is also determined by the
nanoribbon geometry. The bandgap was proven to have the strongest effect on
both the plasmon dispersions and their Landau damping. We have also
demonstrated that for a small electron doping the plasmon dispersions do not
depend on the relative hopping parameter $\alpha$ of the considered $\alpha -
\mathcal{T}_3$ material in the long-wave limit and investigated the conditions
when $\alpha$ becomes an important factor which strongly affects the plasmons.
We believe that our new uncovered electronic and collective properties of
nano-size $\alpha - \mathcal{T}_3$ribbons will find their applications in the
field of modern electronics and nanodevices.
