In the NISQ era, implementing practical quantum ma002 chine learning faces challenges due to limited resources. A recent advancement involves a novel single-qubit methodology, allowing for the execution of numerous operations on a single qubit. However, its applicability to complex neural network architectures remains uncertain. This study focuses on graph data, known for its intricate structure and inherent challenges for machine learning. Utilizing the quantum walk, we introduce a graph embedding approach to reduce network parameters, resulting in the design of a novel architecture called the single-qubit Quantum Graph Neural Network (sQGNN). Experimental assessments, including simulations, demonstrate the adaptability of sQGNNs to graphs of varying dimensions and compositions. The resource-efficient nature of sQGNNs is showcased through practical executions on quantum computers, using only one qubit. This capability highlights the efficient utilization of limited qubit resources, paving the way for expansive quantum graph neural networks on resource-constrained Variational Quantum Circuits (VQCs). This breakthrough opens up new possibilities for practical applications within the current NISQ regime.