Graph neural network (GNN) models are increasingly being used for the classification of electroencephalography (EEG) data. However, GNN-based diagnosis of neurological disorders, such as Alzheimer's disease (AD), remains a relatively unexplored area of research. Previous studies have relied on functional connectivity methods to infer brain graph structures and used simple GNN architectures for the diagnosis of AD. In this work, we propose a novel adaptive gated graph convolutional network (AGGCN) that can provide explainable predictions. AGGCN adaptively learns graph structures by combining convolution-based node feature enhancement with a correlation-based measure of power spectral density similarity. Furthermore, the gated graph convolution can dynamically weigh the contribution of various spatial scales. The proposed model achieves high accuracy in both eyes-closed and eyes-open conditions, indicating the stability of learned representations. Finally, we demonstrate that the proposed AGGCN model generates consistent explanations of its predictions that might be relevant for further study of AD-related alterations of brain networks.