The linear independence of the edge-connectivity index to other first-, second-, and third-generation topological indices is demonstrated by using principal component analysis for octane isomers. Most of the topological indices are loaded in one factor, while the edge connectivity is loaded in another independent factor. The edge-connectivity index does not produce linear correlations (R ≤ 0.7) with any of the almost 40 topological indices studied. This index produced the best single-variable quantitative structure−property relationship (QSPR) models for five of the seven physicochemical properties of octanes studied. It is concluded that the edge connectivity is an independent index containing important structural information to be used in QSPR/QSAR (QSAR = quantitative structure−activity relationship) studies.

The atom-bond connectivity index (ABC), a novel graph theoretical invariant, based on the connectivity between atoms and bonds in a molecule, is proposed. This structure-descriptor is computed from the vertex and edge degrees, but in contrast to the original connectivity index of Randic ABC does not reflect the extent of branching of the molecule. ABC is used to describe the heats of formation of alkanes, resulting in a good quantitative structure-property relationship (QSPR) model ( r = 0.9970). The model is interpreted in such a manner that the intercept and slope of the regression equation have a physical

Higher order analogues of the Wiener number are defined, providing rather precise regression models for a number of physico-chemical properties of alkanes, which certainly are much better than the analogous models based solely on the Wiener number. The new structure descriptors are defined on the basis of the Wiener–Hosoya polynomial HG(x): the kth extended Wiener index kW is equal to the kth derivative of HG(x), evaluated at x=1, in which case 1W is just the original Wiener number.

The Harary H number and MTI descriptor were expressed by means of a matrix-vector-matrix multiplication procedure. Two new series of topological indices analogous to them were defined. MTI-like descriptors were expressed as sums of other topological indices defined in the literature. Both series of molecular descriptors were used to describe boiling points of 21 alkanes. Very significant improvements relative to Harary number were obtained by using some of the novel Harary-like numbers. Almost all MTI-like descriptors show better correlations with boiling point than the original MTI. All the molecular descriptors defined here are expressed in terms of local vertex properties, such as degrees and distance numbers.

The Wiener number is expressed in terms of vector-matrix multiplication procedure by using distance matrix and unit vectors. This approach is extended to the definition of a new series of graph theoretical invariants based on distance and vertex-adjacency matrices. The performance of these topological indices, which include the Zagreb group indices, for the description of boiling points of alkanes is analyzed. It is shown that the Wiener index is one of the best descriptors that can be defined by this procedure. However, some improvements to it are obtained by using some of the novel graph invariants. One of the invariants based on vertex-adjacency matrix represents a very significant improvement relative to Zagreb group indices. The structural selectivity of all descriptors is also analyzed.