Predicting vehicle trajectories is crucial for ensuring automated vehicle operation efficiency and safety, particularly on congested multi-lane highways. In such dynamic environments, a vehicle's motion is determined by its historical behaviors as well as interactions with surrounding vehicles. These intricate interactions arise from unpredictable motion patterns, leading to a wide range of driving behaviors that warrant in-depth investigation. This study presents the Graph-based Interaction-aware Multi-modal Trajectory Prediction (GIMTP) framework, designed to probabilistically predict future vehicle trajectories by effectively capturing these interactions. Within this framework, vehicles' motions are conceptualized as nodes in a time-varying graph, and the traffic interactions are represented by a dynamic adjacency matrix. To holistically capture both spatial and temporal dependencies embedded in this dynamic adjacency matrix, the methodology incorporates the Diffusion Graph Convolutional Network (DGCN), thereby providing a graph embedding of both historical states and future states. Furthermore, we employ a driving intention-specific feature fusion, enabling the adaptive integration of historical and future embeddings for enhanced intention recognition and trajectory prediction. This model gives two-dimensional predictions for each mode of longitudinal and lateral driving behaviors and offers probabilistic future paths with corresponding probabilities, addressing the challenges of complex vehicle interactions and multi-modality of driving behaviors. Validation using real-world trajectory datasets demonstrates the efficiency and potential.