In this paper, we classify all the finite groups $G$ such that the commuting graph $Γ_C(G)$, order-sum graph $Γ_{OS}(G)$ and non-inverse graph $Γ_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite groups $G$ for that, these graphs are minimally connected. We also classify some groups for that the co-prime graph $Γ_{CP}(G)$ has minimal edge connectedness. In final part, we classify all the finite groups $G$ for that co-prime graph $Γ_{CP}(G)$ is minimally connected.