Graph neural networks (GNNs) are now the mainstream method for mining graph-structured data and learning low-dimensional node- and graph-level embeddings to serve downstream tasks. However, limited by the bottleneck of interpretability that deep neural networks present, existing GNNs have ignored the issue of estimating the appropriate number of dimensions for the embeddings. Hence, we propose a novel framework called Minimum Graph Entropy principle-guided Dimension Estimation, i.e. MGEDE, that learns the appropriate embedding dimensions for both node and graph representations. In terms of node-level estimation, a minimum entropy function that counts both structure and attribute entropy, appraises the appropriate number of dimensions. In terms of graph-level estimation, each graph is assigned a customized embedding dimension from a candidate set based on the number of dimensions estimated for the node-level embeddings. Comprehensive experiments with node and graph classification tasks and nine benchmark datasets verify the effectiveness and generalizability of MGEDE.