  A new one-dimensional model is proposed for the low-energy vibrational
quantum dynamics of CH5+ based on the motion of an effective particle confined
to 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 and
are related to combinatorial properties of ${\Gamma}_{60}$. The bipartite
structure of ${\Gamma}_{60}$ gives a simple explanation for curious symmetries
observed in numerically exact variational calculations on CH5+.
