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\nContext. 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.\nAims. 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.\nMethods. Using the DisPerSE cosmic web finder, we identify filamentary structures from the 3D gas particle distribution in 324 simulated regions of 30 h−1 Mpc side from T<sc>HE</sc> T<sc>HREE</sc> H<sc>UNDRED</sc> hydrodynamical simulation at redshifts z = 0, 1, and 2. We estimate the connectivity at various apertures for ∼3000 groups and clusters spanning a mass range from 1013 h−1 M⊙ to 1015 h−1 M⊙. Relationships between connectivity and cluster properties like radius, mass, dynamical state, and hydrostatic mass bias are explored.\nResults. 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 the literature, both from observational and simulated datasets. 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-galaxy-traced filaments in the simulation.\n