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As the Internet of Things system becomes more popular and ubiquitous, it has also gradually entered the consumer electronics field. For example, smart home systems have numerous sensors that monitor the environment and interact with the Internet to provide smart services. A large amount of multivariate time series data generated using sensors can provide services for consumers and identify faulty systems through multivariate time series anomaly detection (MTSAD), which is crucial for maintaining system stability. However, representing the complex relationships among multivariate time series is challenging. Recently, graph neural networks and graph structure learning, which can excellently learn complex time series relationships, have been applied to multivariate time series. However, existing research on graph structure learning only constructs k-Nearest Neighbor (kNN) graphs based on the pair-wise similarity between time series. This generates a quadratic cost and only considers partial relationships among sensors. Accordingly, we propose a lightweight graph structure learning-based multivariate time series anomaly detection (GSLAD), which exploits full graph parameterization to learn the graph structure without pair-wise similarity to overcome the quadratic cost and the limited neighbor relationship. GSLAD exploits diffusion convolutional recurrent neural network (DCRNN) to extract temporal and spatial features. The results from the extensive simulations performed on four public real-world datasets demonstrate that the F1 score improved by an average of 5% with less training time compared to existing state-of-the-art methods.