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Tools available in graph theory have been recently applied to signals recorded from the human brain, where its cognitive functions are linked to topological properties of connectivity networks. In this work, we consider resting-state electroencephalography (EEG) signals recorded from healthy subjects and patients suffering from Alzheimer's disease (AD) in two conditions: eyes-open and eyes-closed. The EEGs are used to construct functional brain networks in which the nodes are EEG sensor locations and edges represent functional connectivity between them. The networks are then tested for a number of neurobiologically relevant graph theory metrics. The analyses show that the network properties are stable across all conventional frequency bands. AD brains in eyes-closed condition show significantly reduced local efficiency and modularity measures ( P < 0.05; Wilcoxon's ranksum test). We then use the network metrics as features for discriminating AD from healthy controls. Three feature selection methods (Genetic Algorithms (GA), Binary Particle Swarm Optimization (BPSO) and Social Impact Theory based Optimization (SITO)) are used to select the best feature set. GA with support vector machines (as classifier) results in an accuracy of 83% in eyes-close beta band. The set of optimal features include edge betweenness centrality, global efficiency, modularity and synchronizability. [ABSTRACT FROM AUTHOR]