Diagnosing and treating dementia, including mild cognitive impairment (MCI), is challenging due to diverse disease types and overlapping symptoms. Early MCI detection is vital as it can precede dementia, yet distinguishing it from later stage dementia is intricate due to subtle symptoms. The primary objective of this study is to adopt a complex network perspective to unravel the underlying pathophysiological mechanisms of dementia-related disorders. Leveraging the extensive availability of electroencephalogram (EEG) data, our study focuses on the meticulous identification and analysis of EEG-based brain functional network (BFNs) associated with dementia-related disorders. To achieve this, we employ the Phase Lag Index (PLI) as a connectivity measure, offering a comprehensive view of neural interactions. To enhance the analytical rigor, we introduce a data-driven threshold selection technique. This innovative approach allows us to compare the topological structures of the formulated BFNs using complex network measures quantitatively and statistically. Furthermore, we harness the power of these BFNs by utilizing them as pre-defined graph inputs for a Graph Convolution Network (GCN-net) based approach. The results demonstrate that graph theory metrics, such as the rich-club coefficient, transitivity, and assortativity coefficients, effectively distinguish between MCI, Alzheimer's disease (AD) and vascular dementia (VD). Furthermore, GCN-net achieves high accuracy (95.07% delta, 80.62% theta) and F1-scores (0.92 delta, 0.67 theta), highlighting the effectiveness of EEG-based BFNs in the analysis of dementia-related disorders.