The pinching-antenna system is a novel flexible-antenna technology, which has the capabilities not only to combat large-scale path loss, but also to reconfigure the antenna array in a flexible manner. The key idea of pinching antennas is to apply small dielectric particles on a waveguide of arbitrary length, so that they can be positioned close to users to avoid significant large-scale path loss. This paper investigates the graph neural network (GNN) enabled transmit design for the joint optimization of antenna placement and power allocation in pinching-antenna systems. We formulate the downlink communication system equipped with pinching antennas as a bipartite graph, and propose a graph attention network (GAT) based model, termed bipartite GAT (BGAT), to solve an energy efficiency (EE) maximization problem. With the tailored readout processes, the BGAT guarantees a feasible solution, which also facilitates the 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.