How can we systematically represent the propagation of information? The propagation structure of fake news has been shown to be an important cue for detecting it; yet, existing propagation-based fake news detection methods have mainly relied on ad hoc topological features, and a unified view of cascade patterns is still lacking. To address this, we study news propagation from a spectral view by connecting graph spectra to propagation-related structural properties through rigorous spectral bounds. We introduce several new bounds and integrate them with existing bounds into a unified spectral representation of information propagation. We then use these spectral bounds for downstream classification and design a discrete structural optimization framework to interpret learned propagation patterns. For efficient optimization, we rely on a first-order perturbation approximation and consider both score-guided and bound-guided objectives. Experiments on real-world data reveal meaningful spectral differences between fake and real news, competitive classification performance, and interpretable evolution trajectories from structural optimization. The findings demonstrate the value of spectral analysis for understanding and modeling information propagation.

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