  The interaction energy for the indirect-exchange or
Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction between magnetic spins
localized on lattice sites of the $\alpha$-${\cal T}_3$ model is calculated
using linear response theory. In this model, the $\texttt{AB}$-honeycomb
lattice structure is supplemented with $\texttt{C}$ atoms at the centers of the
hexagonal lattice. This introduces a parameter $\alpha$ for the ratio of the
hopping integral from hub-to-rim and that around the rim of the hexagonal
lattice. A valley and $\alpha$-dependent retarded Greens function matrix is
used to form the susceptibility. Analytic and numerical results are obtained
for undoped $\alpha$-${\cal T}_3$, when the chemical potential is finite and
also in the presence of an applied magnetic field. We demonstrate the
anisotropy of these results when the magnetic impurities are placed on the
$\texttt{A,B}$ and $\texttt{C}$ sublattice sites. Additionally, comparison of
the behavior of the susceptibility of $\alpha$-${\cal T}_3$ with graphene shows
that there is a phase transition at $\alpha=0$.
