Using real numbers as vertex invariants for third-generation topological indexes

First-generation topological indexes (TI’s) were integer numbers obtained by simple (“bookkeeping”) operations from local vertex invariants (LOVI’s), which were integer numbers. Second-generation TI’s were real numbers obtained via sophisticated (‘structural”) operations from integer LOVI’s. Third-generation TI’s are real numbers based on real-number LOVI’s. In successive generations, there is an increasing correlational abiiity and a decreasing degeneracy of TI’s. Four types of newly developed real-number LOVI’s are reviewed: (i) Information-based LOVI’s obtained from topological distances to all other graph vertexes; (ii) Solutions of linear equation systems obtained from triplets consisting of a matrix (adjacency or distance matrix) and two column vectors; (iii) LOVI’s based on eigenvalues and eigenvectors of the two above matrices; (iv) Regressive distance sums and regressive vertex degrees, which are the corresponding LOVI’s (distance sums or vertex degrees) augmented slightly by all other vertexes, whose contributions decrease with increasing distance. When the LOVI’s are based on topological distances, it is easy to include information on the presence and location of multiple bonds and/or heteroatoms. All LOVI’s are validated by intramolecular comparison within various alkanes, and all TI’s are validated both by intermolecular comparison within series of isomeric alkanes and by correlations with physicochemical properties.