The pinching-antenna system is a novel flexible-antenna technology, capable of both mitigating large-scale path loss and reconfiguring antenna arrays adaptively. Its core principle relies on deploying small dielectric particles along a waveguide of arbitrary length, so that the antennas can be positioned close to users to significantly reduce the impact of large-scale path loss. This letter delves into the graph neural network (GNN) enabled transmit design for the joint optimization of antenna placement and power allocation within pinching-antenna systems. We formulate the downlink pinching-antenna system as a bipartite graph, and propose a graph attention network (GAT)-based model, termed bipartite GAT (BGAT), to address the energy efficiency (EE) maximization problem. With the tailored readout processes, the BGAT ensures feasible solutions, which also facilitates unsupervised training. Numerical results demonstrate the effectiveness of pinching antennas in enhancing the system EE as well as the proposed BGAT in terms of optimality, scalability and computational efficiency.