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A new one-dimensional model is proposed for the low-energy vibrationalquantum dynamics of CH5+ based on the motion of an effective particle confinedto a 60-vertex graph ${\Gamma}_{60}$ with a single edge length parameter.Within this model, the quantum states of CH5+ are obtained in analytic form andare related to combinatorial properties of ${\Gamma}_{60}$. The bipartitestructure of ${\Gamma}_{60}$ gives a simple explanation for curious symmetriesobserved in numerically exact variational calculations on CH5+.

Exactly solvable 1D model explains the low-energy vibrational level structure of protonated methane