Olfaction is not as well-understood as vision or audition, nor technologically addressed. Here, Chemical Graph Theory is shown to connect the vibrational spectrum of an odorant molecule, invoked in the Vibration Theory of Olfaction, to its structure, which is germane to the orthodox Shape Theory. Atomistic simulations yield the Eigen-VAlue (EVA) vibrational pseudo-spectra for 20 odorant molecules grouped into 6 different 'perceptual' classes by odour. The EVA is decomposed into peaks corresponding to different types of vibrational modes. A novel secondary pseudo-spectrum, informed by this physical insight-the Peak-Decomposed EVA (PD-EVA)-has been proposed here. Unsupervised Machine Learning (spectral clustering), applied to the PD-EVA, clusters the odours into different 'physical' (vibrational) classes that match the 'perceptual', and also reveal inherent perceptual subclasses. This establishes a physical basis for vibration-based odour classification, harmonizes the Shape and Vibration theories, and points to vibration-based sensing as a promising path towards a biomimetic electronic nose.