Machine learning is currently the most active interdisciplinary field having numerous applications; additionally, machine-learning techniques are used to research quantum many-body problems. In this study, we first propose neural network quantum states (NNQSs) with general input observables and explore a few related properties, such as the tensor product and local unitary op- eration. Second, we determine the necessary and sufficient conditions for the representability of a general graph state using normalized NNQS. Finally, to quantify the approximation degree of a given pure state, we define the best approximation degree using normalized NNQSs. Furthermore, we observe that some N-qubit states can be represented by a normalized NNQS, such as separable pure states, Bell states and GHZ states.