Accurate anomaly detection in multivariate time series (MTS) relies on the effective modeling of dynamic temporal dependencies and spatial interactions. Many existing methods employ autoencoders or recurrent neural networks (RNNs), which are computationally expensive and often face challenges in capturing long-term dependencies because of vanishing gradient problems. In this paper, we propose a novel framework named Mamba-Register-based Graph Neural Network (MRGNN), to effectively capture both global and local dependencies in time series, enabling the model to better understand the overall sequence structure. This is achieved by introducing a new temporal feature modeling module called Mamba-Register Mechanism (MRM), via integrating the register tokens into the Mamba module. In addition, we replace traditional convolutions with Diffusion Graph Convolution (DGC) for spatio-temporal modeling, where a multi-step diffusion is used to iteratively propagate information across multiple hops in the graph, effectively integrating both global and local spatial information, thereby enhancing the accuracy of the model in capturing complex spatial dependencies. Experiments on four public datasets demonstrate that our proposed framework MRGNN outperforms existing state-of-the-art methods, delivering enhanced anomaly detection performance and improving robustness in handling dynamic spatio-temporal relationships. The code is available at https://github.com/SYLan2019/MRGNN .