  We explore the algebraic structure of the solution space of convex
optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when
the population covariance matrix $\Sigma_x$ has an additional latent graphical
constraint, namely, a latent star topology. In particular, we have shown that
CMTFA can have either a rank $ 1 $ or a rank $ n-1 $ solution and nothing in
between. The special case of a rank $ 1 $ solution, corresponds to the case
where just one latent variable captures all the dependencies among the
observables, giving rise to a star topology. We found explicit conditions for
both rank $ 1 $ and rank $n- 1$ solutions for CMTFA solution of $\Sigma_x$. As
a basic attempt towards building a more general Gaussian tree, we have found a
necessary and a sufficient condition for multiple clusters, each having rank $
1 $ CMTFA solution, to satisfy a minimum probability to combine together to
build a Gaussian tree. To support our analytical findings we have presented
some numerical demonstrating the usefulness of the contributions of our work.
