  Galaxy clusters are located in the densest areas of the universe and are
intricately connected to larger structures through the filamentary network of
the Cosmic Web. In this scenario, matter flows from areas of lower density to
higher density. As a result, the properties of galaxy clusters are deeply
influenced by the filaments that are attached to them, which are quantified by
a parameter known as connectivity. We explore the dependence of gas-traced
filaments connected to galaxy clusters on the mass and dynamical state of the
cluster. Moreover, we evaluate the effectiveness of the cosmic web extraction
procedure from the gas density maps of simulated cluster regions. Using the
DisPerSE cosmic web finder, we identify filamentary structures from 3D gas
particle distribution in 324 simulated regions of $30 \, h^{-1}$ Mpc side from
The Three Hundred hydrodynamical simulation at redshifts z=0, 1, and 2. We
estimate the connectivity at various apertures for $\sim3000$ groups and
clusters spanning a mass range from $10^{13} \, h^{-1} \, M_{\odot}$ to
$10^{15} \, h^{-1} \, M_{\odot}$. Relationships between connectivity and
cluster properties like radius, mass, dynamical state and hydrostatic mass bias
are explored. We show that the connectivity is strongly correlated with the
mass of galaxy clusters, with more massive clusters being on average more
connected. This finding aligns with previous studies in literature, both from
observational and simulated data sets. Additionally, we observe a dependence of
the connectivity on the aperture at which it is estimated. We find that
connectivity decreases with cosmic time, while no dependencies on the dynamical
state and hydrostatic mass bias of the cluster are found. Lastly, we observe a
significant agreement between the connectivity measured from gas-traced and
mock-galaxies-traced filaments in the simulation.
