Objective: Longitudinal neuroimaging studies have demonstrated that adolescence is the crucial developmental epoch of continued brain growth and change. A large number of researchers dedicate to uncovering the mechanisms about brain maturity during adolescence. Motivated by both achievement in graph signal processing and recent evidence that some brain areas act as hubs connecting functionally specialized systems, we proposed an approach to detect these regions from spectral analysis perspective. In particular, as human brain undergoes substantial development throughout adolescence, we addressed the challenge by evaluating the functional network difference among age groups from functional magnetic resonance imaging (fMRI) observations. Methods: We treated these observations as graph signals defined on the parcellated functional brain regions and applied graph Laplacian learning based Fourier Transform (GLFT) to transform the original graph signals into frequency domain. Eigen-analysis was conducted afterwards to study the behavior of the corresponding brain regions, which enables the characterization of brain maturation. Result: We first evaluated our method on the synthetic data and further applied the method to resting and task state fMRI imaging data from Philadelphia Neurodevelopmental Cohort (PNC) dataset, comprised of normally developing adolescents from 8 to 22. The model provided a highest accuracy of 95.69% in distinguishing different adolescence stages. Conclusion: We detected 13 hubs from resting state fMRI and 16 hubs from task state fMRI that are highly related to brain maturation process. Significance: The proposed GLFT method is powerful in extracting the brain connectivity patterns and identifying hub regions with a high prediction power