Alzheimer's disease (AD) is classified as a degenerative and progressive dementia and affects approximately 25 million elderly individuals worldwide. This disorder leads to a reduction in people's productivity and imposes restrictions on their daily lives. Electroencephalography (EEG), in which the electrical brain activity is recorded in the form of time series and analyzed using signal processing techniques, is a well-known neurophysiological AD biomarker. EEG is noninvasive, low-cost, has a high temporal resolution, and provides valuable information about brain dynamics in AD. Here, we present an original approach based on the use of quantile graphs (QGs) for the classification of EEG data. QGs map frequency, amplitude and correlation characteristics of a time series (such as the EEG data of an AD patient) into the topological features of a network. The five network topological measures used here - clustering coefficient, mean jump length, betweenness centrality, modularity and Laplacian Estrada index - showed that the QG method is able to differentiate healthy patients (with eyes open or closed) from patients with AD (with eyes open or closed), and indicate which channels (corresponding to 19 different locations on the patients' scalp) provide the best discriminating power. Furthermore, the joint analysis of delta, theta, alpha and beta wave results indicate that all AD patients under study display clear symptoms of the disease and may have it in its late stage, a diagnosis known a priori and supported by our study.