<p>A graph state is the most general, multiparty entangled quantum state used for many quantum informatic tasks such as quantum computation and quantum communication. There are many types of graph states out of which only few have been studied through photonic as well as atomic systems. In this context, some preliminary work on different types of atomic entangled and hyperentangled graph states such as star graph states, hyperentangled ring graph states and hybrid ring graph states have been recently carried out. In this proposal, we put forward a novel approach to engineer and extensively study these states for quantum communication among various quantum networks. The protocol is purely based on the Atomic Bragg Diffraction (ABD) of two-level neutral atoms through the Fock-field under the fully controllable cavity-QED technique. Furthermore, the engineering of these states through ABD, a longer interaction time regime, guarantees the high enough success probability approaching unity and good fidelities with minimum decoherence risk. The generation of cavity and atomic tree graph states while utilizing a single auxiliary atom have also been carried out. The experimental feasibility for the execution of our proposed work under the realistic experimental context has also been elucidated utilizing data from various experimental setups.</p>