In machine learning and data mining, causal relationship discovery is a critical task. While the state-of-the-art Vector Auto-Regressive Linear Non-Gaussian Acyclic Model (VAR-LiNGAM) method excels in uncovering both instantaneous and time-lagged connections, it entails analyzing multiple VAR matrices, leading to heightened parameter complexity. To address this challenge, we introduce the Causal Graph Process-LiNGAM (CGP-LiNGAM), a novel approach that significantly reduces parameter load by focusing on a single causal graph, a Directed Acyclic Graph (DAG). Leveraging Graph Signal Processing (GSP) techniques, our method interprets causal relations with graph shift invariance and uniqueness. Our experimental results demonstrate the superiority and robustness of CGP-LiNGAM, particularly in high-noise environments. Moreover, we showcase its real-world applicability in studying brain connectivity during sleep, underlining its compatibility with previous sleep-related neuroscientific research.