In real-world applications, spectral Graph Neural Networks (GNNs) are powerful tools for processing diverse types of graphs. However, a single GNN often struggles to handle different graph types-such as homogeneous and heterogeneous graphs-simultaneously. This challenge has led to the manual design of GNNs tailored to specific graph types, but these approaches are limited by the high cost of labor and the constraints of expert knowledge, which cannot keep up with the rapid growth of graph data. To overcome these challenges, we propose AutoSGNN, an automated framework for discovering propagation mechanisms in spectral GNNs. AutoSGNN unifies the search space for spectral GNNs by integrating large language models with evolutionary strategies to automatically generate architectures that adapt to various graph types. Extensive experiments on nine widely-used datasets, encompassing both homophilic and heterophilic graphs, demonstrate that AutoSGNN outperforms state-of-the-art spectral GNNs and graph neural architecture search methods in both performance and efficiency.