We have analyzed in detail the structure of the smallest fullerenes C(n) (20 < or = n < or = 72) obtained in B3LYP/6-31G* calculations. All these systems, except for C60, C70, and C72, necessarily present adjacent pentagons in their structure. For the most stable classical isomers, we have studied the variation of bond distances, HOMO-LUMO gaps and enthalpies of formation with fullerene size. A classification of the bonds in 9 different motifs permits to gain insight in the factors determining the fullerene stability. A simplified model that assumes a fixed energy per bond is able to reproduce the calculated enthalpies of formation and to estimate the energy change associated with each structural motif. An extra stabilization of 30-40 kcal/mol is predicted for the fullerenes showing spherical aromaticity.