Graph representations have been widely used in pattern recognition thanks to their powerful representation formalism and rich theoretical background. A number of errortolerant graph matching algorithms such as graph edit distance have been proposed for computing a distance between two labelled graphs. However, they typically suffer from a high computational complexity, which makes it difficult to apply these matching algorithms in a real scenario. In this paper, we propose an efficient graph distance based on the emerging field of geometric deep learning. Our method employs a message passing neural network to capture the graph structure and learns a metric with a siamese network approach. The performance of the proposed graph distance is validated in two application cases, graph classification and graph retrieval of handwritten words, and shows a promising performance when compared with (approximate) graph edit distance benchmarks.