This work investigates large-scale sketch recognition from a graph-native perspective, where free-hand sketches are directly modeled as structured graphs rather than raster images or stroke sequences. We propose SketchGraphNet, a hybrid graph neural architecture that integrates local message passing with a memory-efficient global attention mechanism, without relying on auxiliary positional or structural encodings. To support systematic evaluation, we construct SketchGraph, a large-scale benchmark comprising 3.44 million graph-structured sketches across 344 categories, with two variants (A and R) to reflect different noise conditions. Each sketch is represented as a spatiotemporal graph with normalized stroke-order attributes. On SketchGraph-A and SketchGraph-R, SketchGraphNet achieves Top-1 accuracies of 83.62% and 87.61%, respectively, under a unified training configuration. MemEffAttn further reduces peak GPU memory by over 40% and training time by more than 30% compared with Performer-based global attention, while maintaining comparable accuracy.