We report on a new de novo peptide sequencing algorithm that uses spectral graph partitioning. In this approach, relationships between m/z peaks are represented by attractive and repulsive springs, and the vibrational modes of the spring system are used to infer information about the peaks (such as "likely b-ion" or "likely y-ion"). We demonstrate the effectiveness of this approach by comparison with other de novo sequencers on test sets of ion-trap and QTOF spectra, including spectra of mixtures of peptides. On all datasets, we outperform the other sequencers. Along with spectral graph theory techniques, the new de novo sequencer EigenMS incorporates another improvement of independent interest: robust statistical methods for recalibration of time-of-flight mass measurements. Robust recalibration greatly outperforms simple least-squares recalibration, achieving about three times the accuracy for one QTOF dataset.