Graphs play a central role in modeling complex relationships in data, yet most graph learning methods falter when faced with cold-start nodes--new nodes lacking initial connections--due to their reliance on adjacency information. To tackle this, we propose SPARC, a groundbreaking framework that introduces a novel approach to graph learning by utilizing generalizable spectral embeddings. With a simple yet powerful enhancement, SPARC empowers state-of-the-art methods to make predictions on cold-start nodes effectively. By eliminating the need for adjacency information during inference and effectively capturing the graph's structure, we make these methods suitable for real-world scenarios where new nodes frequently appear. Experimental results demonstrate that our framework outperforms existing models on cold-start nodes across tasks such as node classification, node clustering, and link prediction. SPARC provides a solution to the cold-start problem, advancing the field of graph learning.