The diagnosis of Attention-Deficit/Hyperactivity Disorder (ADHD) using brain functional connectivity (BFC) relies on analyzing the topological structure of the brain and the connectivity strength between regions of interest (ROIs). Traditional approaches typically involve atlas-based ROI extraction and the generation of brain connectivity matrices based on time-dependent amplitude values of these ROIs. However, atlas-based methods struggle to accommodate subject variability, leading to an oversimplified representation of the brain’s neural architecture. These methods also tend to overlook the nonlinear neural connectivity present in the brain’s curved surfaces, as they rely on Euclidean-based linear connectivity. To address these issues, we propose a data-driven approach to ROI extraction that accounts for individual variability and the complex neural topology of different subjects, resulting in generalized ROIs. Additionally, we introduce a novel brain connectivity approach that captures the interplay between local and global functional connectivity patterns within the brain’s geodesic structure, representing nonlinear neural connectivity more accurately. Our contributions include the use of grouped Dictionary Learning (DL) for data-driven ROI extraction and the integration of conventional and Riemannian graph spectral analysis to model brain connectivity as Composite Graph Spectral Analysis (CGSA). This method significantly advances the identification of spatio-spectral features in brain connectivity, achieving ADHD prediction accuracy of 84.26% ± 0.13% with a one-dimensional convolutional neural network (1D-CNN), highlighting the novelty and effectiveness of our approach.