The descriptive and utility power of linear combinations of special construction of connectivity indices (LCXCI) derived by a trial-and-error procedure from a medium-sized set of eight connectivity indices or from a subset of it has been tested on several properties of different classes of bioorganic and inorganic compounds. Two techniques have been tested to choose the appropriate combination of indices : the forward selection and the complete combinatorial technique. While the latter searches the entire combinatorial space and the first searches only a subspace of it, this last, nevertheless, has many advantages among which to be a good tool for an elementary and direct test for newly defined indices. The modeling of the side-chain volume V of 18 amino acids (AA) is perfectly achieved by a composite index together with a connectivity index, and, while the modeling of pI of 21 amino acids is satisfactorily accomplished by special (0)X(v)-fractional indices that are also rather good descriptors of the melting T-m points of 20 amino acids, the solubility S of 16 and 20 amino acids is nicely described by reciprocal and suprareciprocal connectivity indices, respectively, a description that seems to have nothing in common with the modeling of the same property for 23 purines and pyrimidines (PP) achieved by squared supraconnectivity indices. Nevertheless, the modeling of the solubility of the entire heterogeneous class of n = 43 amino acids, purines, and pyrimidines could be satisfactorily achieved with a set of supracomposite indices based on the chi(t)(v) index mainly. The modeling of the motor octane MON number of 30 alkanes and of the melting points of 17 and 14 alkanes shows how far a minimum set of four connectivity indices can positively replace a larger set of 17 indices, while the modeling of the lattice Delta H-L(phi) enthalpies of 20 metal halides by a mixed set of normal and composite indices introduces and stimulates the problem of the definition of a connectivity model for inorganic compounds. The utility of the given LCXCI is generally rather high as many properties can be satisfactorily modeled by one or just two indices (V, pI, S(AA), S(PP), and Delta H-L(phi)) and it can be enhanced, especially when the modeling requires more than three or four indices, with the introduction of the corresponding orthogonal indices.