Graph classification is a fundamental task in domains ranging from molecular property prediction to materials design. While graph neural networks (GNNs) achieve strong performance by learning expressive representations via message passing, they incur high computational costs, limiting their scalability and deployment on resource-constrained devices. Hyperdimensional Computing (HDC), also known as Vector Symbolic Architectures (VSA), offers a lightweight, brain-inspired alternative, yet existing HDC-based graph methods typically struggle to match the predictive performance of GNNs. In this work, we propose VS-Graph, a vector-symbolic graph learning framework that narrows the gap between the efficiency of HDC and the expressive power of message passing. VS-Graph introduces a Spike Diffusion mechanism for topology-driven node identification and an Associative Message Passing scheme for multi-hop neighborhood aggregation entirely within the high-dimensional vector space. Without gradient-based optimization or backpropagation, our method achieves competitive accuracy with modern GNNs, outperforming the prior HDC baseline by 4-5% on standard benchmarks such as MUTAG and DD. It also matches or exceeds the performance of the GNN baselines on several datasets while accelerating the training by a factor of up to 450x. Furthermore, VS-Graph maintains high accuracy even with the hypervector dimensionality reduced to D=128, demonstrating robustness under aggressive dimension compression and paving the way for ultra-efficient execution on edge and neuromorphic hardware.