As one of the fundamental tasks in remote sensing (RS) image understanding, multi-label remote sensing image scene classification (MLRSSC) is attracting increasing research interest. Human beings can easily perform MLRSSC by examining the visual elements contained in the scene and the spatio-topological relationships of these visual elements. However, most of existing methods are limited by only perceiving visual elements but disregarding the spatio-topological relationships of visual elements. With this consideration, this paper proposes a novel deep learning-based MLRSSC framework by combining convolutional neural network (CNN) and graph neural network (GNN), which is termed the MLRSSC-CNN-GNN. Specifically, the CNN is employed to learn the perception ability of visual elements in the scene and generate the high-level appearance features. Based on the trained CNN, one scene graph for each scene is further constructed, where nodes of the graph are represented by superpixel regions of the scene. To fully mine the spatio-topological relationships of the scene graph, the multi-layer-integration graph attention network (GAT) model is proposed to address MLRSSC, where the GAT is one of the latest developments in GNN. Extensive experiments on two public MLRSSC datasets show that the proposed MLRSSC-CNN-GNN can obtain superior performance compared with the state-of-the-art methods.