A general and convenient method is described by which one can obtain the point group, the 13C NMR pattern, and the number of IR- and Raman-active vibrations of any given fullerene structure directly from its molecular graph. The method is based on certain ‘‘topological’’ atomic coordinates, the Cartesian components of which in a topological principal axis system are proportional to the components of three selected eigenvectors of the adjacency matrix of the graph. Its practical utility lies in combination with existing theoretical results, including analytical molecular-orbital rules which predict two distinct families of closed-shell fullerenes and a systematic search which generates all known Cn fullerene graphs. The overall strategy is illustrated in an experimentally relevant application to the 24 distinct isolated-pentagon fullerene isomers of C84.