Summary: Equidistributive eigenvectors of the adjacency matrix of a symmetric graph have equal weight on equivalent vertices. Real and complex coefficients are in general insufficient to achieve this property for all components of a degenerate manifold of an arbitrary graph. Quaternionic vector coefficients are shown to be necessary for equidistributivity of four- and fivefold degeneracies of the dodecahedron. Possible physical realisations of the equidistributivity property in molecular cages are also considered.