State estimation plays a vital role in the stable operation of modern power systems, but it is vulnerable to cyber attacks. False data injection attacks (FDIA), one of the most common cyber attacks, can tamper with measurement data and bypass the bad data detection (BDD) mechanism, leading to incorrect results of power system state estimation (PSSE). This paper presents a detection framework of FDIA for PSSE based on graph edge-conditioned convolutional networks (GECCN), which use topology information, node features and edge features. Through deep graph architecture, the correlation of sample data is effectively mined to establish the mapping relationship between the estimated values of measurements and the actual states of power systems. In addition, the edge-conditioned convolution operation allows processing data sets with different graph structures. Case studies are undertaken on the IEEE 14-bus system under different attack intensities and degrees to evaluate the performance of GECCN. Simulation results show that GECCN has better detection performance than convolutional neural networks, deep neural networks and support vector machine. Moreover, the satisfactory detection performance obtained with the data sets of the IEEE 14-bus, 30-bus and 118-bus systems verifies the effective scalability of GECCN.