  In this work we consider a class of recursively-grown fractal networks
$G_n(t)$, whose topology is controlled by two integer parameters $t$ and $n$.
We first analyse the structural properties of $G_n(t)$ (including fractal
dimension, modularity and clustering coefficient) and then we move to its
transport properties. The latter are studied in terms of first-passage
quantities (including the mean trapping time, the global mean first-passage
time and the Kemeny's constant) and we highlight that their asymptotic behavior
is controlled by network's size and diameter. Remarkably, if we tune $n$ (or,
analogously, $t$) while keeping the network size fixed, as $n$ increases ($t$
decreases) the network gets more and more clustered and modular, while its
diameter is reduced, implying, ultimately, a better transport performance. The
connection between this class of networks and models for polymer architectures
is also discussed.
