  In this paper we continue our recent analysis [K. Haydukivska et al., J. Mol.
Liq., 2021, 328, 115456] of complex molecules with two branching points at both
ends of the linear backbone with $f_1$ and $f_2$ side arms starting from them,
known as the pom-pom polymers. Here, we analyze the asymmetric case, $f_1 \neq
f_2$, by applying both the analytical approach, based on the direct polymer
renormalization, and computer simulations using both dissipative particle
dynamics and Monte Carlo methods. We study the role played by the molecular
asymmetry of average polymer conformations, considering the infinite dilution
regime and good solvent conditions.The quantitative estimates are reported for
the set of universal size and shape characteristics of such molecules and for
their individual branches, all the functions of $f_1$ and $f_2$. In particular,
we evaluate the size ratio of the gyration radii of symmetric and asymmetric
pom-pom topologies with the same molecular weight and quantitatively reveal an
increase of the effective size of a molecule caused by its asymmetry. We also
introduce and analyse the asymmetry factor and estimate the shift of the center
of mass caused by the presence of side stars, which can serve as another
characteristic of the asymmetry of pom-pom structure.
