Learning meaningful representations of free-hand sketches remains a challenging task given the signal sparsity and the high-level abstraction of sketches. Existing techniques have focused on exploiting either the static nature of sketches with convolutional neural networks (CNNs) or the temporal sequential property with recurrent neural networks (RNNs). In this work, we propose a new representation of sketches as multiple sparsely connected graphs. We design a novel graph neural network (GNN), the multigraph transformer (MGT), for learning representations of sketches from multiple graphs, which simultaneously capture global and local geometric stroke structures as well as temporal information. We report extensive numerical experiments on a sketch recognition task to demonstrate the performance of the proposed approach. Particularly, MGT applied on 414k sketches from Google QuickDraw: 1) achieves a small recognition gap to the CNN-based performance upper bound (72.80% versus 74.22%) and infers faster than the CNN competitors and 2) outperforms all RNN-based models by a significant margin. To the best of our knowledge, this is the first work proposing to represent sketches as graphs and apply GNNs for sketch recognition. Code and trained models are available at https://github.com/PengBoXiangShang/multigraph_transformer.