Recently, Graph Foundation Models (GFMs) have attracted increasing attention for their potential to learn unified and generalizable knowledge across diverse graphs, thereby supporting a wide range of graph scenarios. However, unlike natural language and images, graphs lack an intuitive and unified form for perceiving and organizing transferable knowledge. Therefore, despite many initial explorations, a key problem remains unresolved: the transferable cognitive mechanism for graphs. Existing GFMs usually rely on intuitively defined mechanisms to encode transferable graph patterns, such as handcrafted structural templates (e.g., cycles and trees), fixed propagation mechanisms, and predefined random-walk patterns. These mechanisms are often prescriptive and rigid, limiting their ability to flexibly characterize diverse graph patterns across domains. This motivates the exploration of a more flexible, graph-native, and transferable cognitive mechanism. To this end, we analyze graphs from the spectral perspective and propose SPR. SPR decomposes graph information into Chebyshev polynomial bases and learns shared spectral responses, enabling unified yet adaptive cognition of continuously varying spectral patterns across graphs. We further model cross-graph relational geometry to distill recurring pairwise organization from multiple graphs, providing a shared relational reference for transferring such graph cognition across domains. Extensive experiments across diverse graphs and downstream scenarios demonstrate that SPR enables effective and transferable graph cognition.

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