Abstract: Multi-document summarization (MDS) aims at producing a good-quality summary for several related documents. In this paper, we propose a spectral-based hypothesis, which states that the goodness of summary candidate is closely linked to its so-called spectral impact. Here spectral impact considers the perturbation to the dominant eigenvalue of affinity matrix when dropping the summary candidate from the document cluster. The hypothesis is validated by three theoretical perspectives: semantic scaling, propagation dynamics and matrix perturbation. According to the hypothesis, we formulate the MDS task as the combinatorial optimization of spectral impact and propose an accelerated greedy solution based on a surrogate of spectral impact. The evaluation results on various datasets demonstrate: (1) The performance of the summary candidate is positively correlated with its spectral impact, which accords with our hypothesis; (2) Our spectral-based method has a competitive result as compared to state-of-the-art MDS systems.