  Matter distribution around clusters is highly anisotropic from their being
the nodes of the cosmic web. Clusters' shape and the number of filaments they
are connected to, i.e., their connectivity, should reflect their level of
anisotropic matter distribution and must be, in principle, related to their
physical properties. We investigate the influence of the dynamical state and
the formation history on both the morphology and local connectivity of about
2400 groups and clusters of galaxies from the large hydrodynamical simulation
IllustrisTNG at z=0. We find that the mass of groups and clusters mainly
influences the geometry of the matter distribution: massive halos are
significantly more elliptical, and more connected to the cosmic web than
low-mass ones. Beyond the mass-driven effect, ellipticity and connectivity are
correlated and are imprints of the growth rate of groups and clusters. Both
anisotropy measures appear to trace different dynamical states, such that
unrelaxed groups and clusters are more elliptical and more connected than
relaxed ones. This relation between matter anisotropies and dynamical state is
the sign of different accretion histories. Relaxed groups and clusters are
mostly formed long time ago, and slowly accreting matter at the present time.
They are rather spherical and weakly connected to their environment, mostly
because they had enough time to relax and, hence, lost the connection with
their preferential directions of accretion and merging. In contrast,
late-formed unrelaxed objects are highly anisotropic with large connectivities
and ellipticities. These groups and clusters are in formation phase and must be
strongly affected by the infalling of materials from filaments.
